Generalizing with binomial coefficients (bit advanced)

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.000
In the last video,
we figured out

00:00:02.000 --> 00:00:05.590
the probability of getting
exactly three heads when

00:00:05.590 --> 00:00:08.319
we have five flips
of a fair coin.

00:00:08.319 --> 00:00:10.360
What I want to do in this
video is think about it

00:00:10.360 --> 00:00:12.131
in a slightly more general way.

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We're still going to
assume we have a fair coin,

00:00:14.130 --> 00:00:15.270
although we'll
shortly see we don't

00:00:15.270 --> 00:00:16.780
have to make that assumption.

00:00:16.780 --> 00:00:18.320
But what I want to
do is figure out

00:00:18.320 --> 00:00:23.320
the probability
of getting k heads

00:00:23.320 --> 00:00:30.560
in n flips of the fair coin.

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So the first thing
to think about

00:00:36.530 --> 00:00:38.580
is just how many
possibilities there

00:00:38.580 --> 00:00:42.640
are where there's
going to be n flips.

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So literally, the first flip,
second flip, third flip,

00:00:45.330 --> 00:00:47.670
all the way to the n-th flip.

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And this is a fair coin.

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Each of these, there's two
equally likely possibilities.

00:00:52.930 --> 00:00:54.430
So the total number
of possibilities

00:00:54.430 --> 00:00:58.900
is going to be 2 times
2 times 2, n times.

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So this is going to be equal
to 2 to the n-th possibilities.

00:01:06.720 --> 00:01:09.580
Now, let's think about how
many of those equally likely--

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and these are all equally
likely possibilities.

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This is a fair coin.

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Let's think about
how many of those

00:01:13.948 --> 00:01:21.220
equally likely possibilities
involve k heads.

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Well, just like we
did for the case

00:01:22.950 --> 00:01:27.490
where we're thinking
about three heads,

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we say, well, look,
the first of those k

00:01:30.250 --> 00:01:36.220
heads-- how many different
buckets could it fall into?

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Well, the first of
the k heads could

00:01:37.880 --> 00:01:41.810
fall into n different
buckets, right?

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It could be the first
flip, second flip,

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all the way to the n-th flip.

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Then the second
of those k heads--

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and we don't know
exactly how many

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k is-- will have n
minus 1 possibilities.

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And then the third
of those k heads

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will have n minus
2 possibilities,

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since two of the spots
are already taken up.

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And we would do this
until we have essentially

00:02:03.290 --> 00:02:05.570
accounted for all
of the k heads.

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So this will go down all the
way to-- so the number of things

00:02:11.940 --> 00:02:17.470
we're multiplying is going to be
k, one for each of the k heads.

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So this is 1, 2,
3, and then you're

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going to go all the way
to n minus k minus 1.

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And you could try this
out in the case of 5.

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When n was 5 and k was 3, this
resulted in 5 times 4 times--

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and actually, we just
have to go times 3.

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And actually, that was
this term, right over here.

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So I'm doing a case that
is a little bit longer,

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where k is a slightly
larger number.

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Because this right
over here is 5 minus 2.

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That is this one over here.

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Just so not to confuse you,
let me write it like this.

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So you'll have n spots
for that first head, n

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minus 1 spots for
that second head.

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And then you keep going.

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And you're going to have
a total of these k things

00:03:07.970 --> 00:03:09.070
you're multiplying.

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And that last one
is going to have

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n minus k minus 1 possibilities.

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And now, hopefully, it'll
map a little bit better

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to the one where we had
five flips and three heads.

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Because here, there
was five possibilities

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for the first head,
four possibilities

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for the second head.

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And then n is 5.

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5 minus 2-- you get
three possibilities

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for the last head.

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So this actually works.

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This is the number of
spots, or the number of ways

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that we can put
three heads into five

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different possible buckets.

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Now just like the last video, we
don't want to over-count things

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because we don't
care about the order.

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We don't want to differentiate
one ordering of heads.

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And I'm just going to use
these letters to differentiate

00:04:00.680 --> 00:04:01.513
the different heads.

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We don't want to
differentiate this from this,

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heads A, heads B. Or any of
the other orderings of this.

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So what we need to do is we
need to divide this number

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so that we don't count all
of those different orderings.

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We need to divide this
by the different ways

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that you can order k things.

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So if you have k things,
so one thing, two things,

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all the way to k things, how
many ways can you order it?

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Well, the first thing can
be in k different positions.

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Or maybe I'll do it like this.

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Maybe I'll do it Thing 1.

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I'll call it T for Thing--
Thing 1, Thing 2, Thing 3,

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all the way to Thing k.

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How many different
ways can you order it?

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Well, Thing 1 can be in
k different positions.

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Then Thing 2 will be
in k minus 1 positions.

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And then all the way
down to the last one

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is only going to
have one position.

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So this is going to be k
times k minus 1 times k

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minus 2, all the way down to 1.

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And in the example where we
had three heads in five flips,

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this was 3 times 2
all the way down to 1.

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3 times 2 times 1.

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And so is there a simpler
way that we can write this?

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Well, this expression
right over here,

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this is the same
thing as k factorial.

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And if you haven't ever
heard of what a factorial is,

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it's exactly this thing
right over here. k factorial

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literally means k times k
minus 1 times k minus 2,

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all the way down to 1.

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So for example, 2 factorial
is equal to 2 times 1.

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3 factorial is equal
to 3 times 2 times 1.

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4 factorial is equal to 4
times 3 times 2 times 1.

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And it's actually a
fun thing to play with.

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The factorials get large
very, very, very, very, fast.

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So anyway, this
denominator right over here

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can be rewritten as k factorial.

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This right over here can be
rewritten as k factorial.

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And is there any way to
rewrite this numerator

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in terms of factorials?

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So if we were to write
n factorial-- so let

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me see how we can write this.

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If we were to
write n factorial--

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get some real estate over
here-- so n factorial

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would be equal to n times
n minus 1 times n minus 2,

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all the way down to 1.

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That's kind of what we want.

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We just want the
first k terms of it.

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So what if we divide this
by n minus k factorial.

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So let's think about
what that is going to do.

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If we have n minus k factorial,
that is the same thing

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as-- we have to do a little
bit of algebraic manipulation

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right over here-- that is the
same thing as n minus k times n

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minus k minus 1 times n minus k
minus 2, all the way down to 1.

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When you divide these, the
1's going to cancel out.

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And what you may or
may not realize--

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and you could work out
the math-- is everything

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is going to cancel out here
until you're just left with,

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up here, everything from n times
n minus 1 to n minus k minus 1.

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Because this, if
you expand this out,

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or if you distribute
this negative number,

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this is the same thing
as n minus k plus 1.

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So n minus k plus 1
is the integer that's

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one larger than this
right over here.

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So if you were to
divide it out, this

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would cancel with
something up here.

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This would cancel with
something up here.

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This would cancel with
something up here.

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And what you're
going to be left with

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is exactly this thing over here.

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And if you don't believe me,
we can actually try it out.

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So let's think about
what 5 factorial over 5

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minus 3 factorial
is going to be.

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So this is going to be 5
times 4 times 3 times 2 times

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1-- so all of that stuff up
there, all the way down to 1.

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Over-- 5 minus 3 is 2--
over 2 factorial, 2 times 1.

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2 cancels with 2.

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1 cancels with 1.

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You don't even have
to worry about that.

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And you're just left
with 5 times 4 times 3.

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Exactly what we had up
here, 5 times 4 times 3.

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And so in general, if
you wanted to figure out

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the number of ways to stick
two things in five chairs,

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and you don't care
about differentiating

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between those two
things, you're going

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to have this expression
right over here, which

00:09:06.770 --> 00:09:10.440
is the same thing as
this right over here.

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So you're going to
have n factorial

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over n minus k factorial.

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And then you're
going to divide it

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by this expression right
over here, which we already

00:09:20.560 --> 00:09:23.010
said is the same
thing as k factorial.

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So you're also going to
divide it by k factorial.

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And then you have
a generalized way

00:09:28.530 --> 00:09:31.220
of figuring out
the number of ways

00:09:31.220 --> 00:09:39.040
you can stick k things
in n different buckets, k

00:09:39.040 --> 00:09:42.050
heads in n different flips.

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And so another way
of writing-- and this

00:09:43.945 --> 00:09:45.480
is actually a
generalized formula

00:09:45.480 --> 00:09:47.830
for binomial coefficients.

00:09:47.830 --> 00:09:52.050
So another way to write
this is the number of ways,

00:09:52.050 --> 00:09:59.530
given that you have n buckets,
you can put k things in them

00:09:59.530 --> 00:10:01.490
without having to
differentiate it.

00:10:01.490 --> 00:10:05.630
Or another way to think about
it is if you have n buckets,

00:10:05.630 --> 00:10:10.757
or n flips, and you want to
choose k of them to be heads.

00:10:10.757 --> 00:10:12.590
Or you want to choose
k of them in some way,

00:10:12.590 --> 00:10:14.600
but you don't want
to differentiate.

00:10:14.600 --> 00:10:18.850
So all of these are generalized
ways for binomial coefficients.

00:10:18.850 --> 00:10:21.460
So going back to the
original problem,

00:10:21.460 --> 00:10:24.790
what is the probability
of getting k heads

00:10:24.790 --> 00:10:26.780
in n flips of the fair coin?

00:10:26.780 --> 00:10:30.520
Well, there's 2 to the n
equally likely possibilities.

00:10:30.520 --> 00:10:32.240
So let's write this down.

00:10:32.240 --> 00:10:34.710
So the probability-- you have
2 to the n equally likely

00:10:34.710 --> 00:10:36.020
possibilities.

00:10:36.020 --> 00:10:38.410
And how many of those
possibilities result

00:10:38.410 --> 00:10:40.980
in exactly k heads?

00:10:40.980 --> 00:10:43.740
Well, we just figured that
out during this video.

00:10:43.740 --> 00:10:45.810
That's the number
of possibilities.

00:10:45.810 --> 00:10:52.300
So n factorial over k factorial
times n minus k factorial.

00:10:52.300 --> 00:10:56.440
Now, it probably is a OK
idea to memorize this.

00:10:56.440 --> 00:10:59.270
But I'll just tell you frankly,
the only reason why I still

00:10:59.270 --> 00:11:01.395
know how to do this 20
years after first seeing it,

00:11:01.395 --> 00:11:03.728
or whenever I first saw it,
is that I actually just like

00:11:03.728 --> 00:11:05.170
to reason through it every time.

00:11:05.170 --> 00:11:08.950
I like to just reason through,
OK, I've got five flips.

00:11:08.950 --> 00:11:10.514
Three of them need to be heads.

00:11:10.514 --> 00:11:12.930
The first of those heads can
be in five different buckets,

00:11:12.930 --> 00:11:14.080
and the next in four
different buckets,

00:11:14.080 --> 00:11:15.850
the next one in three
different buckets.

00:11:15.850 --> 00:11:17.500
And then, of
course, I don't want

00:11:17.500 --> 00:11:23.250
to differentiate between
all of the different ways

00:11:23.250 --> 00:11:25.640
that I can rearrange
three different things.

00:11:25.640 --> 00:11:30.320
So I have to make sure that
I divide it by 3 factorial,

00:11:30.320 --> 00:11:32.490
by 3 times 2 times 1.

00:11:32.490 --> 00:11:34.010
I want to make sure
that I divide it

00:11:34.010 --> 00:11:35.650
by all of the
different ways that I

00:11:35.650 --> 00:11:38.840
can arrange three
different things.


Getting exactly two heads (combinatorics)

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.840
I'm going to start
with a fair coin,

00:00:02.840 --> 00:00:06.850
and I'm going to
flip it four times.

00:00:10.410 --> 00:00:14.320
And the first question I want to
ask is, what is the probability

00:00:14.320 --> 00:00:22.800
that I get exactly
one head, or heads?

00:00:22.800 --> 00:00:24.960
This is one of those
confusing things, when you're

00:00:24.960 --> 00:00:26.840
talking about what
side of the coin.

00:00:26.840 --> 00:00:29.592
I know I've been not
doing this consistently.

00:00:29.592 --> 00:00:31.300
I'm tempted to say,
if you're saying one,

00:00:31.300 --> 00:00:33.883
it feels like you should do the
singular, which would be head.

00:00:33.883 --> 00:00:36.177
But I've read up a little
bit of it on the internet,

00:00:36.177 --> 00:00:38.260
and it seems like when
you're talking about coins,

00:00:38.260 --> 00:00:40.980
you really should say one
heads, which is a little bit,

00:00:40.980 --> 00:00:42.672
it seems a little
bit difficult for me.

00:00:42.672 --> 00:00:43.880
But I'll try to go with that.

00:00:43.880 --> 00:00:46.770
So what is the probability
of getting exactly one heads?

00:00:46.770 --> 00:00:48.270
And I put that in
quotes to say, you

00:00:48.270 --> 00:00:50.990
know, really, we're just to
talking about one head there.

00:00:50.990 --> 00:00:53.370
But it's called heads when
you're dealing with coins.

00:00:53.370 --> 00:00:55.810
Anyway, I think you get
what I'm talking about.

00:00:55.810 --> 00:00:57.310
And to think about
this, let's think

00:00:57.310 --> 00:01:01.120
about how many different
possible ways we can get four

00:01:01.120 --> 00:01:02.730
flips of a coin.

00:01:02.730 --> 00:01:05.269
So we're going to have one
flip, then another flip, then

00:01:05.269 --> 00:01:07.120
another flip, then another flip.

00:01:07.120 --> 00:01:09.460
And this first flip
has two possibilities.

00:01:09.460 --> 00:01:10.810
It could be heads or tails.

00:01:10.810 --> 00:01:12.790
The second flip has
two possibilities.

00:01:12.790 --> 00:01:14.230
It could be heads or tails.

00:01:14.230 --> 00:01:15.865
The third flip has
two possibilities.

00:01:15.865 --> 00:01:17.800
It could be heads or tails.

00:01:17.800 --> 00:01:19.860
And the fourth flip
has two possibilities.

00:01:19.860 --> 00:01:21.340
It could be heads or tails.

00:01:21.340 --> 00:01:23.250
So you have 2 times
2 times 2 times

00:01:23.250 --> 00:01:26.730
2, which is equal
to 16 possibilities.

00:01:26.730 --> 00:01:37.060
16 possible outcomes when
you flip a coin four times.

00:01:37.060 --> 00:01:40.970
And any one of the possible
outcomes would be 1 of 16.

00:01:40.970 --> 00:01:42.617
So if I wanted to
say, so if I were to

00:01:42.617 --> 00:01:44.200
just say the
probability, and I'm just

00:01:44.200 --> 00:01:45.824
going to not talk
about this one heads,

00:01:45.824 --> 00:01:48.225
if I just take a, just
maybe this thing that

00:01:48.225 --> 00:01:49.350
has three heads right here.

00:01:49.350 --> 00:01:51.000
This exact sequence of events.

00:01:51.000 --> 00:01:53.660
This is the first flip, second
flip, third flip, fourth flip.

00:01:53.660 --> 00:01:55.480
Getting exactly
this, this is exactly

00:01:55.480 --> 00:02:00.080
one out of a possible
of 16 events.

00:02:00.080 --> 00:02:02.020
Now with that out of
the way, let's think

00:02:02.020 --> 00:02:05.850
about how many possibilities,
how many of those 16

00:02:05.850 --> 00:02:09.370
possibilities, involve
getting exactly one heads?

00:02:09.370 --> 00:02:10.509
Well, we could list them.

00:02:10.509 --> 00:02:11.950
You could get your heads.

00:02:11.950 --> 00:02:16.000
So this is equal
to the probability

00:02:16.000 --> 00:02:20.500
of getting the heads
in the first flip,

00:02:20.500 --> 00:02:26.460
plus the probability of getting
the heads in the second flip,

00:02:26.460 --> 00:02:29.520
plus the probability of getting
the heads in the third flip.

00:02:29.520 --> 00:02:30.890
Remember, exactly one heads.

00:02:30.890 --> 00:02:33.870
We're not saying at least
one, exactly one heads.

00:02:33.870 --> 00:02:37.280
So the probability in
the third flip, and then,

00:02:37.280 --> 00:02:40.840
or the possibility that you
get heads in the fourth flip.

00:02:40.840 --> 00:02:43.694
Tails, heads, and tails.

00:02:43.694 --> 00:02:45.360
And we know already
what the probability

00:02:45.360 --> 00:02:47.120
of each of these things are.

00:02:47.120 --> 00:02:50.270
There are 16 possible
events, and each of these

00:02:50.270 --> 00:02:53.710
are one of those
16 possible events.

00:02:53.710 --> 00:03:02.370
So this is going to be 1 over
16, 1 over 16, 1 over 16,

00:03:02.370 --> 00:03:04.401
and 1 over 16.

00:03:04.401 --> 00:03:06.150
And so we're really
saying the probability

00:03:06.150 --> 00:03:08.067
of getting exactly one
heads is the same thing

00:03:08.067 --> 00:03:10.316
as the probability of getting
heads in the first flip,

00:03:10.316 --> 00:03:12.010
or the probability
of getting heads--

00:03:12.010 --> 00:03:14.010
or I should say the
probability of getting heads

00:03:14.010 --> 00:03:16.140
in the first flip, or
heads in the second flip,

00:03:16.140 --> 00:03:18.692
or heads in the third flip,
or heads in the fourth flip.

00:03:18.692 --> 00:03:21.150
And we can add the probabilities
of these different things,

00:03:21.150 --> 00:03:24.150
because they are
mutually exclusive.

00:03:24.150 --> 00:03:26.600
Any two of these things cannot
happen at the same time.

00:03:26.600 --> 00:03:28.521
You have to pick one
of these scenarios.

00:03:28.521 --> 00:03:30.020
And so we can add
the probabilities.

00:03:30.020 --> 00:03:34.360
1/16 plus 1/16 plus
1/16 plus 1/16.

00:03:34.360 --> 00:03:35.500
Did I say that four times?

00:03:35.500 --> 00:03:37.170
Well, assume that I did.

00:03:37.170 --> 00:03:42.810
And so you would get 4/16,
which is equal to 1/4.

00:03:42.810 --> 00:03:44.210
Fair enough.

00:03:44.210 --> 00:03:47.090
Now let's ask a slightly
more interesting question.

00:03:47.090 --> 00:03:50.960
Let's ask ourselves
the probability

00:03:50.960 --> 00:03:52.855
of getting exactly two heads.

00:03:58.510 --> 00:04:01.900
And there's a couple of
ways we can think about it.

00:04:01.900 --> 00:04:04.540
One is just in the
traditional way.

00:04:04.540 --> 00:04:06.340
[? We ?] know the
number of possibilities

00:04:06.340 --> 00:04:08.580
and of those equally
likely possibilities.

00:04:08.580 --> 00:04:10.260
And we can only use
this methodology

00:04:10.260 --> 00:04:11.830
because it's a fair coin.

00:04:11.830 --> 00:04:14.100
So, how many of the
total possibilities

00:04:14.100 --> 00:04:17.649
have two heads of the total of
equally likely possibilities?

00:04:17.649 --> 00:04:20.420
So we know there are 16
equally likely possibilities.

00:04:20.420 --> 00:04:22.680
How many of those
have two heads?

00:04:22.680 --> 00:04:25.610
So I've actually, ahead
of time so we save time,

00:04:25.610 --> 00:04:29.320
I've drawn all of the 16
equally likely possibilities.

00:04:29.320 --> 00:04:31.610
And how many of these
involve two heads?

00:04:31.610 --> 00:04:32.380
Well, let's see.

00:04:32.380 --> 00:04:37.220
This one over here has two
heads, this one over here

00:04:37.220 --> 00:04:41.770
has two heads, this one
over here has two heads.

00:04:41.770 --> 00:04:46.530
Let's see, this one
over here has two heads,

00:04:46.530 --> 00:04:50.000
and this one over
here has two heads.

00:04:50.000 --> 00:04:53.500
And then this one over
here has two heads,

00:04:53.500 --> 00:04:56.490
and I believe we
are done after that.

00:04:56.490 --> 00:04:59.890
So if we count them, one,
two, three, four, five,

00:04:59.890 --> 00:05:03.630
six of the possibilities
have exactly two heads.

00:05:03.630 --> 00:05:08.050
So six of the 16 equally likely
possibilities have two heads.

00:05:08.050 --> 00:05:11.300
So we have a-- what
is this-- a 3/8 chance

00:05:11.300 --> 00:05:13.102
of getting exactly two heads.

00:05:13.102 --> 00:05:15.310
Now that's kind of what
we've been doing in the past,

00:05:15.310 --> 00:05:17.059
but what I want to do
is think about a way

00:05:17.059 --> 00:05:19.630
so we wouldn't have to write
out all the possibilities.

00:05:19.630 --> 00:05:21.550
And the reason why
that's useful is,

00:05:21.550 --> 00:05:23.279
we're only dealing
with four flips now.

00:05:23.279 --> 00:05:24.820
But if we were
dealing with 10 flips,

00:05:24.820 --> 00:05:26.210
there's no way that
we could write out

00:05:26.210 --> 00:05:27.543
all the possibilities like this.

00:05:27.543 --> 00:05:29.935
So we really want a different
way of thinking about it.

00:05:29.935 --> 00:05:31.310
And the different
way of thinking

00:05:31.310 --> 00:05:35.087
about it is, if we're
saying exactly two heads,

00:05:35.087 --> 00:05:36.920
you can imagine we're
having the four flips.

00:05:36.920 --> 00:05:41.630
Flip one, flip two,
flip three, flip four.

00:05:41.630 --> 00:05:45.660
So these are the flips,
or you could say,

00:05:45.660 --> 00:05:47.400
the outcome of the flips.

00:05:47.400 --> 00:05:49.370
And if you're going to
have exactly two heads,

00:05:49.370 --> 00:05:50.870
you could say, well,
look, I'm going

00:05:50.870 --> 00:05:53.010
to have one head in
one of these positions,

00:05:53.010 --> 00:05:55.430
and then one head in
the other position.

00:05:55.430 --> 00:05:58.360
So how many, if I'm
picking the first,

00:05:58.360 --> 00:06:03.960
so I have kind of a heads
one, and I have a heads two.

00:06:03.960 --> 00:06:07.240
And I don't want you to
think that these are somehow

00:06:07.240 --> 00:06:09.940
the heads in the first flip or
the heads in the second flip.

00:06:09.940 --> 00:06:11.620
What I'm saying is
we need two heads.

00:06:11.620 --> 00:06:14.320
We need a total of two
heads in all of our flips.

00:06:14.320 --> 00:06:16.900
And I'm just giving one
of the heads a name,

00:06:16.900 --> 00:06:18.810
and I'm giving the
other head a name.

00:06:18.810 --> 00:06:21.830
And what we're going to see in a
few seconds is that we actually

00:06:21.830 --> 00:06:23.330
don't want to double
count, we don't

00:06:23.330 --> 00:06:25.570
want to count the
situation, we don't

00:06:25.570 --> 00:06:28.090
want to double count this
situation-- heads one,

00:06:28.090 --> 00:06:30.050
heads two, tails, tails.

00:06:30.050 --> 00:06:33.330
And heads two, heads
one, tails, tails.

00:06:33.330 --> 00:06:36.800
For our purposes, these are
the exact same outcomes.

00:06:36.800 --> 00:06:38.490
So we don't want to
double count that.

00:06:38.490 --> 00:06:40.690
And we're going to have
to account for that.

00:06:40.690 --> 00:06:42.750
But if we just think
about it generally,

00:06:42.750 --> 00:06:45.540
how many different spots,
how many different flips can

00:06:45.540 --> 00:06:47.480
that first head show up in?

00:06:47.480 --> 00:06:49.000
Well, there's four
different flips

00:06:49.000 --> 00:06:51.140
that that first head
could show up in.

00:06:51.140 --> 00:06:57.650
So there's four possibilities,
four flips, or four places

00:06:57.650 --> 00:06:59.700
that it could show up in.

00:06:59.700 --> 00:07:02.490
Well, if that first head takes
up one of these four places,

00:07:02.490 --> 00:07:06.550
let's just say that first head
shows up on the third flip,

00:07:06.550 --> 00:07:09.090
then how many different places
can that second head show up

00:07:09.090 --> 00:07:10.120
in?

00:07:10.120 --> 00:07:12.410
Well, if that first head is
in one of the four places,

00:07:12.410 --> 00:07:15.790
then that second head can only
be in three different places.

00:07:15.790 --> 00:07:18.200
So that second
head can only be--

00:07:18.200 --> 00:07:20.000
I'm picking a nice
color here-- can only

00:07:20.000 --> 00:07:24.287
be in three different places.

00:07:24.287 --> 00:07:26.370
And so, you know, it could
be in any one of these.

00:07:26.370 --> 00:07:27.828
It could maybe be
right over there.

00:07:27.828 --> 00:07:29.750
Any one of those three places.

00:07:29.750 --> 00:07:32.350
And so, when you think about
it in terms of the first,

00:07:32.350 --> 00:07:34.460
and I don't want to say
the first head, head one.

00:07:34.460 --> 00:07:35.876
Actually, let me
call it this way.

00:07:35.876 --> 00:07:38.682
Let me call it head A
and head B. That way

00:07:38.682 --> 00:07:40.890
you won't think that I'm
talking about the first flip

00:07:40.890 --> 00:07:42.160
or the second flip.

00:07:42.160 --> 00:07:46.640
So this is head A, and this
right over there is head B.

00:07:46.640 --> 00:07:49.790
So if you had a particular,
I mean, these heads

00:07:49.790 --> 00:07:50.760
are identical.

00:07:50.760 --> 00:07:52.610
These outcomes aren't
different, but the way

00:07:52.610 --> 00:07:54.690
we talk about it
right now, it looks

00:07:54.690 --> 00:07:57.160
like there's four places that
we could get this head in,

00:07:57.160 --> 00:07:59.710
and there's three places where
we could get this head in.

00:07:59.710 --> 00:08:02.060
And so if you were to multiply
all of the different ways

00:08:02.060 --> 00:08:04.740
that you could get, all
of the different scenarios

00:08:04.740 --> 00:08:06.440
where this is in four
different places,

00:08:06.440 --> 00:08:07.940
and then this is
in one of the three

00:08:07.940 --> 00:08:10.500
left over places, you get
12 different scenarios.

00:08:18.480 --> 00:08:20.670
But there would only be
12 different scenarios

00:08:20.670 --> 00:08:24.520
if you viewed this as
being different than this.

00:08:24.520 --> 00:08:27.400
And let me rewrite it with
our new-- So this is head A,

00:08:27.400 --> 00:08:30.310
this is head B,
this is head B, this

00:08:30.310 --> 00:08:33.289
is head A. There would only be
12 different scenarios if you

00:08:33.289 --> 00:08:35.640
viewed these two things as
fundamentally different.

00:08:35.640 --> 00:08:36.559
But we don't.

00:08:36.559 --> 00:08:38.110
We're actually double counting.

00:08:38.110 --> 00:08:40.140
Because we can
always swap these two

00:08:40.140 --> 00:08:42.799
heads and have the
exact same outcome.

00:08:42.799 --> 00:08:45.750
So what you want to do is
actually divide it by two.

00:08:45.750 --> 00:08:48.000
So you want to divide it by
all of the different ways

00:08:48.000 --> 00:08:50.792
that you can swap
two different things.

00:08:50.792 --> 00:08:53.000
If we had three heads here,
you would think about all

00:08:53.000 --> 00:08:55.370
of the different ways you could
swap three different things.

00:08:55.370 --> 00:08:57.953
If you had four heads here, it
would be all the different ways

00:08:57.953 --> 00:08:59.720
you could swap four
different things.

00:08:59.720 --> 00:09:01.890
So there's 12
different scenarios

00:09:01.890 --> 00:09:03.740
if you couldn't
swap them, but you

00:09:03.740 --> 00:09:05.760
want to divide it by all
of the different ways

00:09:05.760 --> 00:09:07.110
that you can swap two things.

00:09:07.110 --> 00:09:10.810
So 12 divided by
2 is equal to 6.

00:09:10.810 --> 00:09:15.420
Six different scenarios,
fundamentally different

00:09:15.420 --> 00:09:17.650
scenarios, considering
that you can swap them.

00:09:20.180 --> 00:09:23.310
If you assume that head A and
head B can be interchangeable.

00:09:23.310 --> 00:09:26.180
But it's a completely
identical outcome for us,

00:09:26.180 --> 00:09:28.390
because they're
really just heads.

00:09:28.390 --> 00:09:30.280
So there's six
different scenarios,

00:09:30.280 --> 00:09:37.000
and we know that there's a total
of 16 equally likely scenarios.

00:09:37.000 --> 00:09:39.670
So we could say that the
probability of getting exactly

00:09:39.670 --> 00:09:43.520
two heads is 6
times, six scenarios

00:09:43.520 --> 00:09:45.300
and-- Or there's
a couple of ways.

00:09:45.300 --> 00:09:47.050
You could say there
are six scenarios that

00:09:47.050 --> 00:09:48.920
give us two heads,
of a possible 16.

00:09:48.920 --> 00:09:52.670
Or you could say there are
six possible scenarios,

00:09:52.670 --> 00:09:55.790
and the probability of each
of those scenarios is 1/16.

00:09:55.790 --> 00:09:59.680
But either way, you'll
get the same answer.


Exactly three heads in five flips

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.330
So let's start again
with a fair coin.

00:00:03.330 --> 00:00:06.060
And this time, instead of
flipping it four times,

00:00:06.060 --> 00:00:08.060
let's flip it five times.

00:00:08.060 --> 00:00:12.677
So five flips of this fair coin.

00:00:12.677 --> 00:00:14.510
And what I want to think
about in this video

00:00:14.510 --> 00:00:21.831
is the probability of
getting exactly three heads.

00:00:21.831 --> 00:00:23.330
And the way I'm
going to think about

00:00:23.330 --> 00:00:26.740
it is, if you have five flips,
how many different equally

00:00:26.740 --> 00:00:28.169
likely possibilities are there?

00:00:28.169 --> 00:00:30.460
So you're going to have the
first flip-- let me draw it

00:00:30.460 --> 00:00:31.736
over here-- first flip.

00:00:31.736 --> 00:00:33.235
And there's two
possibilities there.

00:00:33.235 --> 00:00:34.680
It could be heads or tails.

00:00:34.680 --> 00:00:36.550
Second flip, two
possibilities there.

00:00:36.550 --> 00:00:38.080
Third flip, two possibilities.

00:00:38.080 --> 00:00:39.900
Fourth flip, two possibilities.

00:00:39.900 --> 00:00:41.590
Fifth flip, two possibilities.

00:00:41.590 --> 00:00:45.720
So it's 2 times 2 times
2 times 2 times 2.

00:00:45.720 --> 00:00:47.840
I hope I said that five times.

00:00:47.840 --> 00:00:49.750
Or you could view that as
2 to the fifth power.

00:00:49.750 --> 00:00:52.970
And that is going
to be equal to 32

00:00:52.970 --> 00:00:56.710
equally likely possibilities.

00:00:56.710 --> 00:00:59.510
32-- 2 times 2 is 4, 4 times
2 is 8, 8 times 2 is 16,

00:00:59.510 --> 00:01:02.870
16 times 2 is 32 possibilities.

00:01:02.870 --> 00:01:04.450
And to figure out
this probability,

00:01:04.450 --> 00:01:06.575
we really just have to
figure out how many of those

00:01:06.575 --> 00:01:10.780
possibilities involve
getting three heads.

00:01:10.780 --> 00:01:13.870
We could write out all
the 32 possibilities

00:01:13.870 --> 00:01:15.386
and literally just
count the heads.

00:01:15.386 --> 00:01:17.010
But let's just use
that other technique

00:01:17.010 --> 00:01:19.760
that we just started to
explore in that last video.

00:01:19.760 --> 00:01:21.610
We have five flips here.

00:01:21.610 --> 00:01:26.920
So let me draw the flips--
one, two, three, four, five.

00:01:26.920 --> 00:01:28.800
And we want to have
exactly three heads.

00:01:28.800 --> 00:01:34.100
And I'm going to call
those three heads-- let

00:01:34.100 --> 00:01:39.859
me do it in pink-- heads
A, heads B, heads C,

00:01:39.859 --> 00:01:40.900
just to give them a name.

00:01:40.900 --> 00:01:42.790
Although what we're going
to see later in this video

00:01:42.790 --> 00:01:45.140
is that we don't want to
differentiate between them.

00:01:45.140 --> 00:01:50.270
To us, it makes no difference if
we get this ordering-- heads A,

00:01:50.270 --> 00:01:54.090
heads B, heads C, tails, tails.

00:01:54.090 --> 00:01:57.020
Or if we get this
ordering-- heads A,

00:01:57.020 --> 00:02:01.660
heads C, or heads
B, tails, tails.

00:02:01.660 --> 00:02:03.910
We can't count these as
two different orderings.

00:02:03.910 --> 00:02:05.474
We can only count this as one.

00:02:05.474 --> 00:02:06.890
So what we're going
to do is first

00:02:06.890 --> 00:02:08.681
come up with all of
the different orderings

00:02:08.681 --> 00:02:11.640
if we cared about the
difference between A, B, and C.

00:02:11.640 --> 00:02:14.490
And then we're going to divide
by all of the different ways

00:02:14.490 --> 00:02:17.530
that you can arrange
three different things.

00:02:17.530 --> 00:02:19.980
So how many ways
can we put A, B,

00:02:19.980 --> 00:02:23.900
and C into these five buckets
that we can view as the flips,

00:02:23.900 --> 00:02:26.490
if we cared about A, B, and C ?

00:02:26.490 --> 00:02:30.110
So let's start with A.
If we haven't allocated

00:02:30.110 --> 00:02:33.260
any of these buckets to
any of the heads yet,

00:02:33.260 --> 00:02:35.940
then we could say that A could
be in five different buckets.

00:02:35.940 --> 00:02:39.550
So there's five possibilities
where A could be.

00:02:39.550 --> 00:02:42.440
So let's just say that this
is the one that it goes in,

00:02:42.440 --> 00:02:44.390
although it could be in
any one of these five.

00:02:44.390 --> 00:02:46.692
But if this takes
up one of the five,

00:02:46.692 --> 00:02:48.400
then how many different
possibilities can

00:02:48.400 --> 00:02:50.120
this heads sit in?

00:02:50.120 --> 00:02:52.090
How many different
possibilities are there?

00:02:52.090 --> 00:02:54.530
Well, then there's only going
to be four buckets left.

00:02:54.530 --> 00:02:56.640
So then there's only
four possibilities.

00:02:56.640 --> 00:02:59.480
And so if this was
where heads A goes,

00:02:59.480 --> 00:03:01.500
then heads B could be in
any of the other four.

00:03:01.500 --> 00:03:03.260
If heads A was in
this first one, then

00:03:03.260 --> 00:03:05.054
heads B could have been
in any of the four.

00:03:05.054 --> 00:03:06.470
I'll just do a
particular example.

00:03:06.470 --> 00:03:08.950
Maybe heads B shows
up right there.

00:03:08.950 --> 00:03:10.680
So once we've taken
two of the slots,

00:03:10.680 --> 00:03:13.740
how many spaces do
we have for heads C?

00:03:13.740 --> 00:03:17.611
Well, we only have three
spaces left, then, for heads C.

00:03:17.611 --> 00:03:19.610
And so it could be in any
of these three spaces.

00:03:19.610 --> 00:03:21.200
And just to show a
particular example,

00:03:21.200 --> 00:03:22.550
it would look like that.

00:03:22.550 --> 00:03:26.900
And so if you cared about order,
how many different ways can

00:03:26.900 --> 00:03:30.760
you, out of five
different spaces,

00:03:30.760 --> 00:03:33.160
allocate them to
three different heads?

00:03:33.160 --> 00:03:36.750
You would say it is
5 times 4 times 3.

00:03:36.750 --> 00:03:43.410
5 times 4 is 20, times
3 is equal to 60.

00:03:43.410 --> 00:03:45.760
So you would say there
are 60 different ways

00:03:45.760 --> 00:03:50.250
to arrange heads A, B, and C
in five buckets, or five flips,

00:03:50.250 --> 00:03:52.770
or if these were
people, in five chairs.

00:03:52.770 --> 00:03:55.762
And obviously, there
aren't 60 possibilities

00:03:55.762 --> 00:03:56.720
of getting three heads.

00:03:56.720 --> 00:03:59.920
In fact, there's only 32
equally likely possibilities.

00:03:59.920 --> 00:04:02.550
And the reason why we got
such a big number over here

00:04:02.550 --> 00:04:05.980
is that we are counting this
scenario as being fundamentally

00:04:05.980 --> 00:04:13.220
different than if this was
heads B, heads A, and then

00:04:13.220 --> 00:04:15.310
heads C over here.

00:04:15.310 --> 00:04:17.060
And what we need to
do is say, well, these

00:04:17.060 --> 00:04:18.351
aren't different possibilities.

00:04:18.351 --> 00:04:21.180
We don't have to
overcount for all

00:04:21.180 --> 00:04:22.960
of the different ways
you arrange this.

00:04:22.960 --> 00:04:24.750
And so what we need
to do is divide this

00:04:24.750 --> 00:04:26.320
by all of the
different ways that you

00:04:26.320 --> 00:04:28.830
can arrange three things.

00:04:28.830 --> 00:04:34.060
So if I have three things
that are in three spaces.

00:04:34.060 --> 00:04:37.800
So here I have a heads in
the second flip, third flip,

00:04:37.800 --> 00:04:38.890
and fifth flip.

00:04:38.890 --> 00:04:41.360
If I have three things in
three spaces like this,

00:04:41.360 --> 00:04:43.900
how many ways can
I arrange them?

00:04:43.900 --> 00:04:47.140
And so if I have three
spaces, how many ways can I

00:04:47.140 --> 00:04:51.520
arrange an A, B, and C
in those three spaces?

00:04:51.520 --> 00:04:54.150
Well, A can go
into three spaces.

00:04:54.150 --> 00:04:55.580
It can go into any of the three.

00:04:59.670 --> 00:05:03.230
Then B would have two spaces
left, once A takes one of them.

00:05:03.230 --> 00:05:05.300
And then C would
have one space left,

00:05:05.300 --> 00:05:06.880
once A and B take two of them.

00:05:06.880 --> 00:05:11.050
So there's 3 times 2
times 1 way to arrange

00:05:11.050 --> 00:05:12.310
three different things.

00:05:12.310 --> 00:05:16.180
So 3 times 2 times
1 is equal to 6.

00:05:16.180 --> 00:05:19.480
So the number of possibilities
of getting three heads

00:05:19.480 --> 00:05:24.110
is actually going to be
this 5 times 4 times 3.

00:05:24.110 --> 00:05:26.310
Let me write this
down in another color.

00:05:26.310 --> 00:05:32.550
So the number of possibilities--
let's write "poss" for short--

00:05:32.550 --> 00:05:47.170
is equal to this 5 times 4
times 3 over the number of ways

00:05:47.170 --> 00:05:48.820
that I can rearrange
three things.

00:05:48.820 --> 00:05:52.420
Because we don't want to
overcount for all of these,

00:05:52.420 --> 00:05:55.057
viewing this arrangement
as fundamentally

00:05:55.057 --> 00:05:56.390
different than this arrangement.

00:05:56.390 --> 00:06:00.820
So then we want to
divide it by-- I

00:06:00.820 --> 00:06:04.440
want to do that same orange
color-- dividing it by 3 times

00:06:04.440 --> 00:06:05.360
2 times 1.

00:06:09.740 --> 00:06:18.210
And which gives us, in the
numerator, 120 divided by 6.

00:06:18.210 --> 00:06:20.270
Oh sorry, that's not--
it's 60 divided by 6.

00:06:20.270 --> 00:06:20.845
This is 60.

00:06:20.845 --> 00:06:23.010
5 times 4 times 3 is 60.

00:06:23.010 --> 00:06:27.600
It gives us 60 divided
by 6, which gives us

00:06:27.600 --> 00:06:34.020
10 possibilities that gives
us exactly three heads.

00:06:34.020 --> 00:06:37.080
And that's of 32 equally
likely possibilities.

00:06:37.080 --> 00:06:39.460
So the probability of getting
exactly three heads-- well,

00:06:39.460 --> 00:06:42.980
you get exactly three
heads in 10 of the 32

00:06:42.980 --> 00:06:44.790
equally likely possibilities.

00:06:44.790 --> 00:06:50.990
So you have a 5/16
chance of that happening.


Alternate proof to induction for integer sum

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.630
In the last video, we
proved that the sum

00:00:02.630 --> 00:00:06.310
of all of the positive
integers up to and including n

00:00:06.310 --> 00:00:11.350
can be expressed as n
times n plus 1 over 2.

00:00:11.350 --> 00:00:12.959
And we proved that by induction.

00:00:12.959 --> 00:00:14.750
What I want to do in
this video is show you

00:00:14.750 --> 00:00:16.730
that there's actually a
simpler proof for that.

00:00:16.730 --> 00:00:19.490
But it's not by
induction, so it wouldn't

00:00:19.490 --> 00:00:20.630
be included in that video.

00:00:20.630 --> 00:00:22.005
But I'll show you
that it exists,

00:00:22.005 --> 00:00:24.960
just so you know that induction
isn't the only way to prove it.

00:00:24.960 --> 00:00:28.630
So we define that
function S of n

00:00:28.630 --> 00:00:32.150
as the sum of all of the
positive integers up to

00:00:32.150 --> 00:00:34.080
and including n.

00:00:34.080 --> 00:00:36.050
So this is equal
to, by definition,

00:00:36.050 --> 00:00:43.900
1 plus 2 plus 3 plus, all the
way to plus n minus 1 plus n.

00:00:43.900 --> 00:00:47.320
So it's the sum of all of the
integers up to and including n.

00:00:47.320 --> 00:00:49.230
This is how we're defining it.

00:00:49.230 --> 00:00:50.530
Well, we can rewrite it again.

00:00:50.530 --> 00:00:53.840
We can say that the sum,
S of n-- we could just

00:00:53.840 --> 00:00:55.840
rewrite this same thing,
but we could rewrite it

00:00:55.840 --> 00:00:56.714
in a different order.

00:00:56.714 --> 00:01:01.400
We could say that this is the
same thing as n plus n minus 1

00:01:01.400 --> 00:01:09.990
plus n minus 2 plus, all the
way down to plus 2 plus 1.

00:01:09.990 --> 00:01:12.040
Now, what does this do for us?

00:01:12.040 --> 00:01:14.480
Well, we can actually
add these two rows.

00:01:14.480 --> 00:01:17.150
If we add S of n
plus S of n, we're

00:01:17.150 --> 00:01:21.090
going to get 2 times
this sum, so we're just

00:01:21.090 --> 00:01:22.400
adding on the left.

00:01:22.400 --> 00:01:25.060
And then we can also
add on the right.

00:01:25.060 --> 00:01:27.630
So we're just adding this sum
twice, but what's interesting

00:01:27.630 --> 00:01:28.984
is how we're going to add it.

00:01:28.984 --> 00:01:31.150
We're going to add this
term to this term, this term

00:01:31.150 --> 00:01:33.066
to this term, because
we're really just trying

00:01:33.066 --> 00:01:34.180
to add these two things.

00:01:34.180 --> 00:01:36.330
And we can pick any way
we want to add them.

00:01:36.330 --> 00:01:43.320
So 1 plus n is going
to be n plus 1.

00:01:43.320 --> 00:01:46.810
And then we're going to add--
let me do it in pink-- 2

00:01:46.810 --> 00:01:49.010
plus n minus 1.

00:01:49.010 --> 00:01:50.637
So what's 2 plus n minus 1?

00:01:50.637 --> 00:01:51.720
Let me write it over here.

00:01:51.720 --> 00:01:54.460
2 plus n minus 1.

00:01:54.460 --> 00:01:57.110
It's the same thing
as 2 plus n minus 1,

00:01:57.110 --> 00:01:59.400
which is the same
thing as n plus 1.

00:01:59.400 --> 00:02:01.240
2 minus 1 is just 1.

00:02:01.240 --> 00:02:04.470
So this is also
going to be n plus 1.

00:02:04.470 --> 00:02:10.190
And then this term over here,
3 plus n minus 2, or n minus 2

00:02:10.190 --> 00:02:11.260
plus 3.

00:02:11.260 --> 00:02:14.550
Once again, that's
going to be n plus 1.

00:02:14.550 --> 00:02:16.300
And you're going to
do that for every term

00:02:16.300 --> 00:02:20.640
all the way until you get
over here, n minus 1 plus 2.

00:02:20.640 --> 00:02:25.400
That's also going
to be n plus 1.

00:02:25.400 --> 00:02:29.130
And then finally, you have
n plus 1 right over here.

00:02:29.130 --> 00:02:32.390
Plus n plus 1.

00:02:32.390 --> 00:02:34.940
So what's this whole
sum going to be?

00:02:34.940 --> 00:02:38.330
Well, how many of these
n plus 1's do we have?

00:02:38.330 --> 00:02:40.870
Well, we have n of
them, for every term

00:02:40.870 --> 00:02:42.030
in each of these sums.

00:02:42.030 --> 00:02:45.540
So this is 1, 2, 3,
count all the way to n.

00:02:45.540 --> 00:02:46.980
You have n of these terms.

00:02:46.980 --> 00:02:49.910
So you have n n plus 1's.

00:02:49.910 --> 00:02:52.750
So if you add something
to itself n times,

00:02:52.750 --> 00:02:56.300
or if you have something
n times right over here,

00:02:56.300 --> 00:03:01.190
this is exactly equivalent
to n times n plus 1.

00:03:01.190 --> 00:03:06.920
So 2 times that sum of
all the positive integers

00:03:06.920 --> 00:03:08.660
up to and including
n is going to be

00:03:08.660 --> 00:03:10.930
equal to n times n plus 1.

00:03:10.930 --> 00:03:12.930
So if you divide
both sides by 2,

00:03:12.930 --> 00:03:15.430
we get an expression
for the sum.

00:03:15.430 --> 00:03:19.400
So the sum of all the positive
integers up to and including n

00:03:19.400 --> 00:03:24.330
is going to be equal to
n times n plus 1 over 2.

00:03:24.330 --> 00:03:27.730
So here was a proof where we
didn't have to use induction.

00:03:27.730 --> 00:03:30.941
It's really kind of a
pure algebraic proof.


Proof by induction

WEBVTT
Kind: captions
Language: en

00:00:00.933 --> 00:00:05.133
I'm going to define a function S of n

00:00:05.133 --> 00:00:06.759
and I'm going to define it as

00:00:06.759 --> 00:00:16.754
the sum of all positive integers

00:00:16.754 --> 00:00:22.221
including N.

00:00:22.221 --> 00:00:24.995
And so the domain of this function is really all

00:00:24.995 --> 00:00:28.533
positive integers - N has to be a positive integer.

00:00:28.533 --> 00:00:30.467
And so we can try this out with a few things, we can

00:00:30.467 --> 00:00:35.200
take S of 3, this is going to be equal to 1 plus

00:00:35.200 --> 00:00:41.071
2 plus 3, which is equal to 6. We could take S of

00:00:41.071 --> 00:00:46.610
4, which is going to be 1 plus 2 plus 3 plus 4,

00:00:46.610 --> 00:00:51.225
which is going to be equal to 10.

00:00:51.225 --> 00:00:57.441
Now what I want to do in this video is prove to you that

00:00:57.441 --> 00:01:02.200
I can write this as a function of N, that the sum of all

00:01:02.200 --> 00:01:04.200
positive integers up to and including N is equal to

00:01:04.200 --> 00:01:11.400
n times n plus one, all of that over 2.

00:01:11.400 --> 00:01:16.821
And the way I'm going to prove it to you is by induction.

00:01:16.821 --> 00:01:23.785
Proof by induction.

00:01:23.785 --> 00:01:27.867
The way you do a proof by induction is first, you prove

00:01:27.867 --> 00:01:36.871
the base case. This is what we need to prove.

00:01:36.871 --> 00:01:41.933
We're going to first prove it for 1 - that will be our base case.

00:01:41.933 --> 00:01:47.400
And then we're going to do the induction step, which

00:01:47.400 --> 00:01:51.000
is essentially saying "If we assume it works for some

00:01:51.000 --> 00:01:55.513
positive integer K", then we can prove it's going to work

00:01:55.513 --> 00:01:59.467
for the next positive integer, for example K + 1.

00:01:59.467 --> 00:02:01.995
And the reason why this works is - Let's say that

00:02:01.995 --> 00:02:10.754
we prove both of these. So the base case we're going to prove it for 1.

00:02:10.769 --> 00:02:13.800
But it doesn't always have to be 1.

00:02:13.800 --> 00:02:16.908
Your statement might be true for everything above 55.

00:02:16.908 --> 00:02:18.667
Or everything above some threshold.

00:02:18.667 --> 00:02:21.533
But in this case, we are saying this is true for all positive integers.

00:02:21.533 --> 00:02:24.333
Our base case is going to be 1.

00:02:24.333 --> 00:02:34.533
Then in our induction step, we are going to prove that if you assume that this thing is true,

00:02:34.533 --> 00:02:37.333
for sum of k.

00:02:37.333 --> 00:02:42.533
If we assume that then it is going to be true for sum of k + 1.

00:02:42.533 --> 00:02:48.333
And the reason why this is all you have to do to prove this for all positive integers

00:02:48.333 --> 00:02:50.000
it's just imagine:

00:02:50.000 --> 00:02:52.000
Let's think about all of the positive integers right over here.

00:02:52.000 --> 00:02:57.467
1, 2, 3, 4, 5, 6 you could just keep going on forever.

00:02:57.467 --> 00:03:00.200
So we are going to prove it for 1.

00:03:00.200 --> 00:03:02.600
We are going to prove that this formula right over here,

00:03:02.600 --> 00:03:07.067
this expression right over here applies for the case of 1, when n is 1.

00:03:07.067 --> 00:03:12.867
And then we are going to prove that if we know it is true for any given k that is true for the next one

00:03:12.867 --> 00:03:16.333
So if we know it is true for 1 in our base case then the second step,

00:03:16.333 --> 00:03:19.067
this induction step must be true for 2 then.

00:03:19.067 --> 00:03:22.867
Because we proved generally if it's true for k it's going to be true for k + 1.

00:03:22.867 --> 00:03:25.733
When it's true for 2, then it must be true for 3,

00:03:25.733 --> 00:03:29.600
because we have proved it, when it's true for k, it's true for k + 1.

00:03:29.600 --> 00:03:31.667
So if it's true for 2 it's true for 3.

00:03:31.667 --> 00:03:34.067
and if it's true for 3 it's going to be true for 4.

00:03:34.067 --> 00:03:37.667
You can just keep going on and on forever, which means it's true for everything.

00:03:37.667 --> 00:03:43.533
Now spoken in generalaties let's actually prove this by induction.

00:03:43.533 --> 00:03:51.800
So let's take the sum of, let's do this function on 1.

00:03:51.800 --> 00:03:54.533
that is just going to be the sum of all positive integers

00:03:54.533 --> 00:03:56.800
including 1 is just literally going to be 1.

00:03:56.800 --> 00:03:58.400
We've just added all of them, it is just 1.

00:03:58.400 --> 00:04:02.000
There is no other positive integer up to and including 1.

00:04:02.000 --> 00:04:04.400
And now we can prove that this is the same thing as

00:04:04.400 --> 00:04:09.600
1 times 1 plus 1 all of that over 2.

00:04:09.600 --> 00:04:14.533
1 plus 1 is 2, 2 divided by 2 is 1, 1 times 1 is 1.

00:04:14.533 --> 00:04:17.533
So this formula right over here, this expression

00:04:17.533 --> 00:04:23.267
it worked for 1, so we have proved our base case.

00:04:23.267 --> 00:04:24.867
we have proven it for 1.

00:04:24.867 --> 00:04:31.385
Now what I want to do is I want to assume that it works for some number k.

00:04:31.385 --> 00:04:41.467
So I will assume true for some number k.

00:04:41.467 --> 00:04:48.000
So i'm going to assume that for some number k, that this function at k is going to be equal to

00:04:48.000 --> 00:04:52.133
k times k plus 1 over 2.

00:04:52.133 --> 00:04:54.200
So I'm just assuming this is true for that.

00:04:54.200 --> 00:05:01.400
Now what I want to do is think about what happens when I try to find this function for k + 1.

00:05:01.400 --> 00:05:03.133
This is what I'm assuming.

00:05:03.133 --> 00:05:05.133
I'm assuming I know this.

00:05:05.133 --> 00:05:07.200
Now let's try to do it for k + 1.

00:05:07.200 --> 00:05:14.323
So what is the sum of all of the integers up to and including k + 1?

00:05:14.323 --> 00:05:20.800
Well this is going to be 1 plus 2 plus 3 plus all the way up to k.

00:05:20.800 --> 00:05:23.667
Plus k plus 1.

00:05:23.667 --> 00:05:26.933
Right? this is the sum of everything up to and including k plus 1.

00:05:26.933 --> 00:05:30.333
Well we are assuming that we know what this already is.

00:05:30.333 --> 00:05:32.600
We are assuming that we already have a formula for this.

00:05:32.600 --> 00:05:39.133
We are assuming that this is going to simplify to k times k plus 1 over 2.

00:05:39.133 --> 00:05:41.000
We are assuming that we know that.

00:05:41.000 --> 00:05:44.733
So we will just take this part and we will add it to k plus 1.

00:05:44.733 --> 00:05:50.190
so we'll add it to k plus 1 over here.

00:05:50.190 --> 00:05:55.467
And if you find the common denominator, the common denominator is going to be 2.

00:05:55.467 --> 00:05:58.667
So this is going to be equal to...

00:05:58.667 --> 00:06:00.600
I'll write the part in magenta first.

00:06:00.600 --> 00:06:10.667
so this is k times k plus 1 over 2 plus 2 times k plus 1 over 2.

00:06:10.667 --> 00:06:13.533
This thing in blue is the same thing as that thing in blue.

00:06:13.533 --> 00:06:16.533
The 2's would cancel out, I'd just wrote it this way so I have a common denominator.

00:06:16.533 --> 00:06:20.615
So this is going to be equal to...

00:06:20.615 --> 00:06:25.800
We have a common denominator of 2 and I'll write this in a different colour here.

00:06:25.800 --> 00:06:32.400
So we are going to have k times k plus 1 plus 2 times k plus 1.

00:06:32.400 --> 00:06:36.867
Now at this step right over here you can factor out a k plus 1.

00:06:36.867 --> 00:06:39.267
Both of these terms are divisible by k + 1.

00:06:39.267 --> 00:06:41.000
So let's factor this out.

00:06:41.000 --> 00:06:46.400
So if you factor out a k + 1, you get k plus 1 times

00:06:46.400 --> 00:06:50.867
refractoring out over here, if you factor out k + 1 you'd just have a k.

00:06:50.867 --> 00:06:54.252
Over here if you factor out k + 1 you would just have a 2.

00:06:54.252 --> 00:06:55.225
Let me colour code those.

00:06:55.225 --> 00:06:56.800
So you would know what I'm doing.

00:06:56.800 --> 00:07:06.667
So this 2 is this 2 right over there and this k is this k right over there.

00:07:06.667 --> 00:07:08.533
We factored it out.

00:07:08.533 --> 00:07:12.533
These k+1's we factored out is this k+1 over there.

00:07:12.533 --> 00:07:17.267
And it's going to be all of this over 2.

00:07:17.267 --> 00:07:21.533
Now, we can rewrite this. This is the same thing.

00:07:21.533 --> 00:07:23.333
This is equal to.

00:07:23.333 --> 00:07:31.867
This is the same thing as k plus 1, that's this part right over here.

00:07:31.867 --> 00:07:37.200
Times k plus 1 plus 1.

00:07:37.200 --> 00:07:39.800
Right? This is clearly the same thing as k + 2.

00:07:39.800 --> 00:07:43.692
All of that over 2.

00:07:43.692 --> 00:07:46.141
Why is this interesting to us?

00:07:46.141 --> 00:07:47.538
Well we have just proven it.

00:07:47.538 --> 00:07:56.467
If we assume that this is true and if we use that assumption we get

00:07:56.467 --> 00:08:05.769
that the sum of all positive integers up to and including k + 1 is equal to k + 1 times k + 1 + 1 over

00:08:05.769 --> 00:08:06.415
2.

00:08:06.415 --> 00:08:13.133
We are actually showing that the original formula applies to k+1 as well.

00:08:13.133 --> 00:08:20.867
If you would take k + 1 and put it in for n you got exactly the result that we got over here.

00:08:20.867 --> 00:08:24.200
So we showed , we proved our base case.

00:08:24.200 --> 00:08:29.533
This expression worked for the sum for all of positive integers up to and including 1.

00:08:29.533 --> 00:08:34.867
And it also works if we assume that it works for everything up to k.

00:08:34.867 --> 00:08:39.533
Or if we assume it works for integer k it also works for the integer k plus 1.

00:08:39.533 --> 00:08:42.467
And we are done. That is our proof by induction.

00:08:42.467 --> 00:08:45.000
That proves to us that it works for all positive integers.

00:08:45.000 --> 00:08:46.400
Why is that?

00:08:46.400 --> 00:08:51.200
We have proven it for 1 and we have proven it that if it works for some integer

00:08:51.200 --> 00:08:56.385
it will work for the next integer.

00:08:56.385 --> 00:08:59.477
So if you assume it worked for 1 then it can work for 2.

00:08:59.477 --> 00:09:03.267
Well we have already proven that it works for 1 so we can assume it works for 1.

00:09:03.267 --> 00:09:05.067
So it definitely will work for 2.

00:09:05.067 --> 00:09:06.800
So we get 2 checked.

00:09:06.800 --> 00:09:10.333
But since we can assume it works for 2 we can now assume it works for 3.

00:09:10.333 --> 00:09:14.000
Well if it works for 3 well then we have proven it works for 4.

00:09:14.000 --> 00:09:16.600
You see how this induction step is kinda like a domino,

00:09:16.600 --> 99:59:59.999
it cascades and we can go on and on forever so it works for all positive integers.


Probability without equally likely events

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.110
So far, we've been dealing
with one way of thinking

00:00:03.110 --> 00:00:06.170
about probability, and
that was the probability

00:00:06.170 --> 00:00:10.310
of A occurring is the
number of events that

00:00:10.310 --> 00:00:20.470
satisfy A over all of the
equally likely events.

00:00:26.350 --> 00:00:30.800
And this is all of the
equally likely events.

00:00:30.800 --> 00:00:34.360
And so in the case
of a fair coin,

00:00:34.360 --> 00:00:37.200
the probability of heads--
well, it's a fair coin.

00:00:37.200 --> 00:00:39.920
So there's two
equally likely events,

00:00:39.920 --> 00:00:42.630
and we're saying one of
them satisfies being heads.

00:00:42.630 --> 00:00:45.230
So there's a 1/2 chance
of you having a heads.

00:00:45.230 --> 00:00:48.100
The same thing for tails.

00:00:48.100 --> 00:00:50.980
If you took a die, and
you said the probability

00:00:50.980 --> 00:00:55.060
of getting an even number
when you roll the die.

00:00:55.060 --> 00:00:57.680
Well, there's six
equally likely events,

00:00:57.680 --> 00:00:59.660
and there's three even
numbers you could get.

00:00:59.660 --> 00:01:02.020
You could get 2, a 4, or a 6.

00:01:02.020 --> 00:01:04.010
So there's three even numbers.

00:01:04.010 --> 00:01:08.219
So once again, you have a
1/2 chance of that happening.

00:01:08.219 --> 00:01:09.760
And this is a really
good model where

00:01:09.760 --> 00:01:12.650
you have equally likely
events happening.

00:01:12.650 --> 00:01:15.204
Now I'm going to change
things up a little bit.

00:01:15.204 --> 00:01:17.620
So I'm going to draw a line
here because this was just one

00:01:17.620 --> 00:01:19.036
way of thinking
about probability.

00:01:19.036 --> 00:01:21.040
Now we're going to
introduce another one that's

00:01:21.040 --> 00:01:24.910
more helpful when we can't think
about equally likely events.

00:01:24.910 --> 00:01:28.770
And in particular, I'm going
to set up an unfair coin.

00:01:28.770 --> 00:01:32.670
So this right over here is
going to be my unfair coin.

00:01:32.670 --> 00:01:35.270
So that is my coin.

00:01:35.270 --> 00:01:37.460
Well, I could draw the coin.

00:01:37.460 --> 00:01:38.830
So it's a gold coin this time.

00:01:38.830 --> 00:01:39.570
It is unfair.

00:01:39.570 --> 00:01:43.031
One side of that coin is a
little heavier than the other,

00:01:43.031 --> 00:01:44.530
even though it's
meant to look fair.

00:01:44.530 --> 00:01:48.190
So it still has that picture
of some president or something

00:01:48.190 --> 00:01:49.710
on one side of it.

00:01:49.710 --> 00:01:51.460
So this is the head side.

00:01:51.460 --> 00:01:54.160
This is heads, and then,
obviously, on the back,

00:01:54.160 --> 00:01:55.870
you have tails.

00:01:55.870 --> 00:01:58.470
But as I mentioned,
this is an unfair coin.

00:01:58.470 --> 00:02:00.450
And I'm going to make
it interesting statement

00:02:00.450 --> 00:02:02.700
about this unfair coin and
one that really doesn't fit

00:02:02.700 --> 00:02:05.544
into the mold that
I set up over here,

00:02:05.544 --> 00:02:07.460
and this interesting
statement is that we have

00:02:07.460 --> 00:02:10.620
more than a 50/50
chance of getting heads

00:02:10.620 --> 00:02:13.520
or more than a 50% chance
or more than a 1/2 chance

00:02:13.520 --> 00:02:14.580
of getting heads.

00:02:14.580 --> 00:02:17.350
I'm going to say that the
probability of getting heads

00:02:17.350 --> 00:02:21.380
for this coin right
over here is 60%.

00:02:21.380 --> 00:02:24.110
Or another way to
say it, it's 0.6.

00:02:24.110 --> 00:02:26.950
Or another way to say
it, it is 6 out of 10.

00:02:26.950 --> 00:02:31.000
Or another way to
say it, it is 3/5.

00:02:31.000 --> 00:02:32.750
And this might make
intuitive sense to you

00:02:32.750 --> 00:02:33.800
and hopefully it
does a little bit,

00:02:33.800 --> 00:02:34.770
but I want you to
realize that this

00:02:34.770 --> 00:02:37.420
is fundamentally different
than what we were saying before

00:02:37.420 --> 00:02:40.340
because now we can't say that
there are two equally likely

00:02:40.340 --> 00:02:40.850
events.

00:02:40.850 --> 00:02:42.174
There are two possible events.

00:02:42.174 --> 00:02:43.590
You can either get
heads or tails.

00:02:43.590 --> 00:02:45.860
We're assuming that the
coin won't fall on its edge.

00:02:45.860 --> 00:02:47.020
That's impossible.

00:02:47.020 --> 00:02:50.120
So you're either going
to get heads or tails,

00:02:50.120 --> 00:02:54.070
but they're not
equally likely anymore.

00:02:54.070 --> 00:02:56.300
So we really can't do
this kind of counting

00:02:56.300 --> 00:02:58.180
the number of events
that satisfy something

00:02:58.180 --> 00:03:00.210
over all of the possible events.

00:03:00.210 --> 00:03:02.980
In this situation, in order
to visualize the probability,

00:03:02.980 --> 00:03:06.070
we have to kind of take what's
called a "frequentist approach"

00:03:06.070 --> 00:03:09.280
or think about it in terms
of frequency probability.

00:03:09.280 --> 00:03:13.500
And the way to conceptualize
a 60% of getting heads

00:03:13.500 --> 00:03:17.510
is to think, if we had a
super large number of trials,

00:03:17.510 --> 00:03:23.110
if we were to just flip
this coin a gazillion times,

00:03:23.110 --> 00:03:27.770
we would expect that 60% of
those would come up heads.

00:03:27.770 --> 00:03:30.570
It's unclear how I
determined that this is 60%.

00:03:30.570 --> 00:03:32.350
Maybe I ran a
computer simulation.

00:03:32.350 --> 00:03:34.850
Maybe I know exactly all
of the physics of this,

00:03:34.850 --> 00:03:38.870
and I could completely model how
it's going to fall every time.

00:03:38.870 --> 00:03:42.350
Or maybe I've actually
just run a ton of trials.

00:03:42.350 --> 00:03:44.760
I've flipped the coin a
million times, and I said,

00:03:44.760 --> 00:03:49.220
wow, 60% of those, 600,000
of those, came up heads.

00:03:49.220 --> 00:03:51.780
And then, we could make a
similar statement about tails.

00:03:51.780 --> 00:03:53.320
So if the probability
of heads is

00:03:53.320 --> 00:03:56.030
60%, the probability
of tails-- well,

00:03:56.030 --> 00:03:58.870
there's only two
possibilities, heads or tails.

00:03:58.870 --> 00:04:01.270
So if I say the probability
of heads or tails,

00:04:01.270 --> 00:04:03.819
it's going to be equal
to 1 because you're

00:04:03.819 --> 00:04:05.360
going to get one of
those two things.

00:04:05.360 --> 00:04:08.770
You have 100% chance of
getting a heads or a tails,

00:04:08.770 --> 00:04:10.780
and these are mutually
exclusive events.

00:04:10.780 --> 00:04:13.500
You can't have both of them.

00:04:13.500 --> 00:04:14.990
The probability
of tails is going

00:04:14.990 --> 00:04:21.529
to be 100% minus the
probability of getting heads,

00:04:21.529 --> 00:04:24.030
and this, of course, is 60%.

00:04:24.030 --> 00:04:30.050
So it's 100% minus 60%, or
40%, or as a decimal, 0.4,

00:04:30.050 --> 00:04:35.220
or as a fraction, 4/10, or as
a simplified fraction, 2/5.

00:04:35.220 --> 00:04:37.560
So, once again,
this probability is

00:04:37.560 --> 00:04:40.270
saying-- we can't say
equally likely events.

00:04:40.270 --> 00:04:41.645
We could say that,
if we're going

00:04:41.645 --> 00:04:43.970
to do a gazillion of
these, we would expect,

00:04:43.970 --> 00:04:46.790
as we get more and more and more
trials, more and more flips,

00:04:46.790 --> 00:04:48.715
40% of those would be heads.

00:04:48.715 --> 00:04:50.590
Now, with that out of
the way, let's actually

00:04:50.590 --> 00:04:52.610
do some problems with this.

00:04:52.610 --> 00:04:55.140
So let's think about the
probability of getting heads

00:04:55.140 --> 00:04:58.720
on our first flip and
heads on our second flip.

00:04:58.720 --> 00:05:01.020
So, once again, these
are independent events.

00:05:01.020 --> 00:05:02.659
The point has no memory.

00:05:02.659 --> 00:05:04.450
Regardless of what I
got on the first flip,

00:05:04.450 --> 00:05:06.090
I have an equal chance
of getting heads

00:05:06.090 --> 00:05:06.930
on the second flip.

00:05:06.930 --> 00:05:09.530
It doesn't matter if I got
heads or tails on the first.

00:05:09.530 --> 00:05:13.170
So this is the probability of
heads on the first flip times

00:05:13.170 --> 00:05:15.980
the probability of heads
on the second flip,

00:05:15.980 --> 00:05:16.860
and we already know.

00:05:16.860 --> 00:05:22.605
The probability of heads on
any flip is going to be 60%.

00:05:22.605 --> 00:05:23.730
I'll write it as a decimal.

00:05:23.730 --> 00:05:29.360
It makes the math a little
bit easier, 0.6, 0.6,

00:05:29.360 --> 00:05:30.796
and we can just multiply.

00:05:30.796 --> 00:05:31.920
I'll do it right over here.

00:05:31.920 --> 00:05:37.200
So this is 0.6 times 0.6.

00:05:37.200 --> 00:05:40.060
Now, it's always good
to do a reality check.

00:05:40.060 --> 00:05:44.530
One way to think about it
is I'm taking 6/10 of 6/10,

00:05:44.530 --> 00:05:47.170
so it should be a little
bit more than half of 6/10

00:05:47.170 --> 00:05:50.067
or probably a little
bit more than 3/10.

00:05:50.067 --> 00:05:51.650
And we've explain
this in detail where

00:05:51.650 --> 00:05:53.060
we talk about
multiplying decimals,

00:05:53.060 --> 00:05:54.934
but we essentially just
multiply the numbers,

00:05:54.934 --> 00:05:56.740
not thinking about
the decimals at first.

00:05:56.740 --> 00:05:58.696
6 times 6 is 36.

00:05:58.696 --> 00:06:00.320
And then you count
the number of digits

00:06:00.320 --> 00:06:01.820
we have to the right
of the decimal.

00:06:01.820 --> 00:06:04.110
We have one, two to the
right of the decimal.

00:06:04.110 --> 00:06:05.735
So we're going to
have two to the right

00:06:05.735 --> 00:06:07.280
of the decimal in our answer.

00:06:07.280 --> 00:06:10.010
So it is 0.36, and
that makes sense.

00:06:10.010 --> 00:06:12.100
We're taking 60% of 0.6.

00:06:12.100 --> 00:06:15.380
We're taking 0.6 of 0.6, a
little bit more than half

00:06:15.380 --> 00:06:16.390
of 0.6.

00:06:16.390 --> 00:06:18.630
And, once again, it's a
little bit more than 0.3.

00:06:18.630 --> 00:06:20.350
So this also makes sense.

00:06:20.350 --> 00:06:21.702
So it's 0.36.

00:06:21.702 --> 00:06:23.160
Or another way to
think about it is

00:06:23.160 --> 00:06:26.570
there's a 36%
probability that we

00:06:26.570 --> 00:06:30.070
get two heads in a row,
given this unfair coin.

00:06:30.070 --> 00:06:32.380
Remember, if it was a fair
coin, it would be 1/2 times

00:06:32.380 --> 00:06:36.350
1/2, which is 1/4, which
is 25%, and it makes sense

00:06:36.350 --> 00:06:38.027
that this is more than that.

00:06:38.027 --> 00:06:40.110
Now, let's think about a
slightly more complicated

00:06:40.110 --> 00:06:40.610
example.

00:06:44.654 --> 00:06:47.100
Let's say the probability
of getting a tails

00:06:47.100 --> 00:06:51.160
on the first flip, getting
a heads on the second flip,

00:06:51.160 --> 00:06:54.163
and then getting a
tails-- I'm going

00:06:54.163 --> 00:06:56.620
to do this in a new
color-- and then getting

00:06:56.620 --> 00:06:59.400
a tails on the third flip.

00:06:59.400 --> 00:07:02.220
So this is going to be equal
to the probability of getting

00:07:02.220 --> 00:07:04.870
a tails on the first flip
because these are all

00:07:04.870 --> 00:07:07.287
independent events.

00:07:07.287 --> 00:07:09.370
If you know that you had
a tail on the first flip,

00:07:09.370 --> 00:07:10.828
that doesn't affect
the probability

00:07:10.828 --> 00:07:12.760
of getting a heads
on the second flip.

00:07:12.760 --> 00:07:15.540
So times the probability
of getting a heads

00:07:15.540 --> 00:07:17.640
on the second flip,
and then that's

00:07:17.640 --> 00:07:20.560
times the probability of getting
a tails on the third flip.

00:07:23.510 --> 00:07:25.970
And the probability of getting
a tails on any flip we know

00:07:25.970 --> 00:07:27.750
is 0.4.

00:07:27.750 --> 00:07:35.150
The probability of getting
a heads on any flip is 0.6,

00:07:35.150 --> 00:07:37.870
and then the probability of
getting tails on any flip

00:07:37.870 --> 00:07:42.060
is 0.4.

00:07:42.060 --> 00:07:45.210
And so, once again, we
can just multiply these.

00:07:45.210 --> 00:07:52.777
So 0.4 times 0.6.

00:07:52.777 --> 00:07:55.110
There's actually a couple of
ways we can think about it.

00:07:55.110 --> 00:07:57.910
Well, we could literally say,
look, we're multiplying 4 times

00:07:57.910 --> 00:08:00.550
6 times 4, and then
we have three numbers

00:08:00.550 --> 00:08:01.790
behind the decimal point.

00:08:01.790 --> 00:08:02.800
So let's do it that way.

00:08:02.800 --> 00:08:05.980
4 times 6 is 24.

00:08:05.980 --> 00:08:10.400
24 times 4 is 96.

00:08:10.400 --> 00:08:16.130
So we write a 96,
but remember, we

00:08:16.130 --> 00:08:19.100
have three numbers
behind the decimal point.

00:08:19.100 --> 00:08:22.897
So it's one to the right
of the decimal there,

00:08:22.897 --> 00:08:24.480
one to the right of
the decimal there,

00:08:24.480 --> 00:08:25.770
one to the right
of decimal there.

00:08:25.770 --> 00:08:26.790
So three to the right.

00:08:26.790 --> 00:08:29.740
So we need three to the right
of the decimal in our answer.

00:08:29.740 --> 00:08:33.070
So one, two-- we need one more
to the right of the decimal.

00:08:33.070 --> 00:08:36.730
So our answer is 0.096.

00:08:36.730 --> 00:08:40.400
Or another way to think about it
is-- write an equal sign here--

00:08:40.400 --> 00:08:43.630
this is equal to a 9.6% chance.

00:08:43.630 --> 00:08:45.680
So there's a little bit
less than 10% chance,

00:08:45.680 --> 00:08:48.130
or a little bit less
than 1 in 10 chance,

00:08:48.130 --> 00:08:50.330
of, when we flip this
coin three times,

00:08:50.330 --> 00:08:52.680
us getting exactly a
tails on the first flip,

00:08:52.680 --> 00:08:58.150
a heads on the second flip,
and a tails on the third flip.


Coin flipping probability

WEBVTT
Kind: captions
Language: en

00:00:00.562 --> 00:00:02.770
Now let's start to do some
more interesting problems.

00:00:02.770 --> 00:00:05.061
And one of these things that
you'll find in probability

00:00:05.061 --> 00:00:07.969
is that you can always do
a more interesting problem.

00:00:07.969 --> 00:00:09.510
So now I'm going to
think about-- I'm

00:00:09.510 --> 00:00:11.010
going to take a
fair coin, and I'm

00:00:11.010 --> 00:00:13.240
going to flip it three times.

00:00:13.240 --> 00:00:16.100
And I want to find the
probability of at least

00:00:16.100 --> 00:00:24.450
one head out of the three flips.

00:00:24.450 --> 00:00:26.130
So the easiest way
to think about this

00:00:26.130 --> 00:00:28.880
is how many equally likely
possibilities there are.

00:00:28.880 --> 00:00:33.170
In the last video, we saw
if we flip a coin 3 times,

00:00:33.170 --> 00:00:35.000
there's 8 possibilities.

00:00:35.000 --> 00:00:37.680
For the first flip,
there's 2 possibilities.

00:00:37.680 --> 00:00:39.620
Second flip, there's
2 possibilities.

00:00:39.620 --> 00:00:41.810
And in the third flip,
there are 2 possibilities.

00:00:41.810 --> 00:00:46.730
So 2 times 2 times 2-- there are
8 equally likely possibilities

00:00:46.730 --> 00:00:50.090
if I'm flipping a coin 3 times.

00:00:50.090 --> 00:00:51.630
Now how many of
those possibilities

00:00:51.630 --> 00:00:53.300
have at least 1 head?

00:00:53.300 --> 00:00:55.240
Well, we drew all the
possibilities over here.

00:00:55.240 --> 00:00:56.990
So we just have to
count how many of these

00:00:56.990 --> 00:00:58.090
have at least 1 head.

00:00:58.090 --> 00:01:05.269
So that's 1, 2, 3, 4, 5 5, 6, 7.

00:01:05.269 --> 00:01:08.060
So 7 of these have at
least 1 head in them.

00:01:08.060 --> 00:01:09.390
And this last one does not.

00:01:09.390 --> 00:01:13.520
So 7 of the 8 have
at least 1 head.

00:01:13.520 --> 00:01:15.189
Now you're probably
thinking, OK, Sal.

00:01:15.189 --> 00:01:16.730
You were able to do
it by writing out

00:01:16.730 --> 00:01:17.840
all of the possibilities.

00:01:17.840 --> 00:01:19.256
But that would be
really hard if I

00:01:19.256 --> 00:01:21.730
said at least one
head out of 20 flips.

00:01:21.730 --> 00:01:23.900
This had worked well
because I only had 3 flips.

00:01:23.900 --> 00:01:25.710
Let me make it clear,
this is in 3 flips.

00:01:30.516 --> 00:01:31.890
This would have
been a lot harder

00:01:31.890 --> 00:01:34.970
to do or more time consuming
to do if I had 20 flips.

00:01:34.970 --> 00:01:36.790
Is there some shortcut here?

00:01:36.790 --> 00:01:38.920
Is there some other
way to think about it?

00:01:38.920 --> 00:01:41.765
And you couldn't just do
it in some simple way.

00:01:41.765 --> 00:01:43.390
You can't just say,
oh, the probability

00:01:43.390 --> 00:01:47.760
of heads times the probability
of heads, because if you got

00:01:47.760 --> 00:01:49.200
heads the first
time, then now you

00:01:49.200 --> 00:01:50.839
don't have to get heads anymore.

00:01:50.839 --> 00:01:52.880
Or you could get heads
again-- you don't have to.

00:01:52.880 --> 00:01:55.750
So it becomes a little
bit more complicated.

00:01:55.750 --> 00:01:58.030
But there is an easy way
to think about it where

00:01:58.030 --> 00:02:00.171
you could use this
methodology right over here.

00:02:00.171 --> 00:02:01.920
You'll actually see
this on a lot of exams

00:02:01.920 --> 00:02:04.129
where they make it seem
like a harder problem,

00:02:04.129 --> 00:02:06.670
but if you just think about in
the right way, all of a sudden

00:02:06.670 --> 00:02:08.340
it becomes simpler.

00:02:08.340 --> 00:02:11.980
One way to think about it is
the probability of at least

00:02:11.980 --> 00:02:15.140
1 head in 3 flips
is the same thing--

00:02:15.140 --> 00:02:22.250
this is the same thing--
as the probability of not

00:02:22.250 --> 00:02:27.040
getting all tails, right?

00:02:27.040 --> 00:02:30.450
If we got all tails, then we
don't have at least 1 head.

00:02:30.450 --> 00:02:32.810
So these two things
are equivalent.

00:02:32.810 --> 00:02:35.560
The probability of getting
at least 1 head in 3 flips

00:02:35.560 --> 00:02:37.380
is the same thing
as the probability

00:02:37.380 --> 00:02:40.810
of not getting all
tails in 3 flips.

00:02:46.010 --> 00:02:49.420
So what's the probability
of not getting all tails?

00:02:49.420 --> 00:02:53.000
Well, that's going to be 1
minus the probability of getting

00:02:53.000 --> 00:02:54.290
all tails.

00:02:54.290 --> 00:02:56.790
The probability of getting
all tails, since it's 3

00:02:56.790 --> 00:03:01.790
flips, it's the probability
of tails, tails, and tails.

00:03:01.790 --> 00:03:03.870
Because any of the
other situations

00:03:03.870 --> 00:03:05.780
are going to have at
least 1 head in them.

00:03:05.780 --> 00:03:07.529
And that's all of the
other possibilities,

00:03:07.529 --> 00:03:09.760
and then this is the only
other leftover possibility.

00:03:09.760 --> 00:03:12.180
If you add them together,
you're going to get 1.

00:03:12.180 --> 00:03:16.040
Let me write it this way.

00:03:16.040 --> 00:03:18.110
Let me write it a
new color just so you

00:03:18.110 --> 00:03:20.100
see where this is coming from.

00:03:20.100 --> 00:03:28.520
The probability of not all
tails plus the probability

00:03:28.520 --> 00:03:33.820
of all tails-- well, this
is essentially exhaustive.

00:03:33.820 --> 00:03:35.990
This is all of the
possible circumstances.

00:03:35.990 --> 00:03:38.830
So your chances of getting
either not all tails

00:03:38.830 --> 00:03:41.310
or all tails-- and these
are mutually exclusive,

00:03:41.310 --> 00:03:44.350
so we can add them.

00:03:44.350 --> 00:03:49.860
The probability of not all
tails or, just to be clear

00:03:49.860 --> 00:03:52.170
what we're doing, the
probability of not all tails

00:03:52.170 --> 00:04:01.370
or the probability of all tails
is going to be equal to one.

00:04:01.370 --> 00:04:02.890
These are mutually exclusive.

00:04:02.890 --> 00:04:05.250
You're either going to
have not all tails, which

00:04:05.250 --> 00:04:06.280
means a head shows up.

00:04:06.280 --> 00:04:07.810
Or you're going
to have all tails.

00:04:07.810 --> 00:04:10.310
But you can't have both
of these things happening.

00:04:10.310 --> 00:04:11.860
And since they're
mutually exclusive

00:04:11.860 --> 00:04:13.568
and you're saying the
probability of this

00:04:13.568 --> 00:04:16.269
or this happening, you could
add their probabilities.

00:04:16.269 --> 00:04:21.570
And this is essentially
all of the possible events.

00:04:21.570 --> 00:04:24.600
So this is essentially,
if you combine these,

00:04:24.600 --> 00:04:26.920
this is the probability of
any of the events happening.

00:04:26.920 --> 00:04:30.560
And that's going to
be a 1 or 100% chance.

00:04:30.560 --> 00:04:32.560
So another way to think
about is the probability

00:04:32.560 --> 00:04:35.270
of not all tails
is going to be 1

00:04:35.270 --> 00:04:37.300
minus the probability
of all tails.

00:04:37.300 --> 00:04:39.460
So that's what we
did right over here.

00:04:39.460 --> 00:04:43.370
And the probability of all
tails is pretty straightforward.

00:04:43.370 --> 00:04:44.990
That's the probability
of it's going

00:04:44.990 --> 00:04:47.370
to be 1/2, because you have
a 1/2 chance of getting

00:04:47.370 --> 00:04:50.106
a tails on the
first flip, times--

00:04:50.106 --> 00:04:52.480
let me write it here, so we
can have it a little clearer.

00:04:52.480 --> 00:04:56.900
So this is going to be 1 minus
the probability of getting

00:04:56.900 --> 00:04:57.779
all tails.

00:04:57.779 --> 00:04:59.570
You will have a 1/2
chance of getting tails

00:04:59.570 --> 00:05:01.030
on the first flip,
and then you're

00:05:01.030 --> 00:05:03.580
going to have to get another
tails on the second flip,

00:05:03.580 --> 00:05:05.080
and then you're
going to have to get

00:05:05.080 --> 00:05:06.650
another tails on the third flip.

00:05:06.650 --> 00:05:10.030
And then 1/2 times
1/2 times 1/2.

00:05:10.030 --> 00:05:12.670
This is going to be 1/8.

00:05:12.670 --> 00:05:16.300
And then 1 minus
1/8 or 8/8 minus 1/8

00:05:16.300 --> 00:05:20.960
is going to be equal to 7/8.

00:05:20.960 --> 00:05:23.300
So we can apply that to
a problem that is harder

00:05:23.300 --> 00:05:25.100
to do than writing
all of the scenarios

00:05:25.100 --> 00:05:26.475
like we did in
the first problem.

00:05:32.600 --> 00:05:37.360
Let's say we have 10 flips,
the probability of at least one

00:05:37.360 --> 00:05:50.130
head in 10 flips-- well,
we use the same idea.

00:05:50.130 --> 00:05:56.770
This is going to be equal
to the probability of not

00:05:56.770 --> 00:06:01.285
all tails in 10 flips.

00:06:03.561 --> 00:06:05.060
So we're just saying
the probability

00:06:05.060 --> 00:06:07.600
of not getting all of the
flips going to be tail.

00:06:07.600 --> 00:06:11.850
All of the flips is tails--
not all tails in 10 flips.

00:06:11.850 --> 00:06:16.690
And this is going to be 1 minus
the probability of flipping

00:06:16.690 --> 00:06:18.780
tails 10 times.

00:06:18.780 --> 00:06:21.325
So it's 1 minus
10 tails in a row.

00:06:26.590 --> 00:06:29.720
And so this is going to be equal
to this part right over here.

00:06:29.720 --> 00:06:30.620
Let me write this.

00:06:30.620 --> 00:06:32.462
So this is going to be this one.

00:06:32.462 --> 00:06:33.420
Let me just rewrite it.

00:06:33.420 --> 00:06:38.670
This is equal to 1 minus-- and
this part is going to be, well,

00:06:38.670 --> 00:06:40.440
one tail, another tail.

00:06:40.440 --> 00:06:41.390
So it's 1/2 times 1/2.

00:06:41.390 --> 00:06:42.806
And I'm going to
do this 10 times.

00:06:45.670 --> 00:06:47.790
Let me write this
a little neater.

00:06:47.790 --> 00:06:56.970
1/2-- so that's 5,
6, 7, 8, 9, and 10.

00:06:56.970 --> 00:07:00.190
And so we really just have to--
the numerator is going to be 1.

00:07:00.190 --> 00:07:03.640
So this is going to be 1.

00:07:03.640 --> 00:07:05.210
This is going to be equal to 1.

00:07:05.210 --> 00:07:07.490
Let me do it in that
same color of green.

00:07:07.490 --> 00:07:10.210
This is going to be equal
to 1 minus-- our numerator,

00:07:10.210 --> 00:07:12.630
you just have 1 times
itself 10 times.

00:07:12.630 --> 00:07:13.920
So that's 1.

00:07:13.920 --> 00:07:18.620
And then on the denominator,
you have 2 times 2 is 4.

00:07:18.620 --> 00:07:32.330
4 times 2 is 8, 16, 32, 64, 128,
256, 512, 1,024-- over 1,024.

00:07:32.330 --> 00:07:42.260
This is the exact same thing
as 1 is 1024 over 1024 minus 1

00:07:42.260 --> 00:07:55.150
over 1024, which is equal
to 1,023 over 1,024.

00:07:55.150 --> 00:07:56.600
We have a common
denominator here.

00:07:56.600 --> 00:08:03.830
So 1,000-- I'm doing that
same blue-- over 1,024.

00:08:03.830 --> 00:08:07.424
So if you flip a coin 10
times in a row-- a fair coin--

00:08:07.424 --> 00:08:09.340
you're probability of
getting at least 1 heads

00:08:09.340 --> 00:08:11.450
in that 10 flips is pretty high.

00:08:11.450 --> 00:08:14.260
It's 1,023 over 1,024.

00:08:14.260 --> 00:08:16.419
And you can get a calculator
out to figure that out

00:08:16.419 --> 00:08:17.460
in terms of a percentage.

00:08:17.460 --> 00:08:20.080
Actually, let me just
do that just for fun.

00:08:20.080 --> 00:08:27.710
So if we have 1,023 divided
by 1,024 that gives us-- you

00:08:27.710 --> 00:08:30.690
have a 99.9% chance
that you're going

00:08:30.690 --> 00:08:33.770
to have at least one heads.

00:08:33.770 --> 00:08:35.330
So this is if we round.

00:08:35.330 --> 00:08:40.034
This is equal to 99.9% chance.

00:08:40.034 --> 00:08:41.159
And I rounded a little bit.

00:08:41.159 --> 00:08:43.909
It's actually slightly, even
slightly, higher than that.

00:08:43.909 --> 00:08:46.580
And this is a pretty powerful
tool or a pretty powerful way

00:08:46.580 --> 00:08:48.660
to think about it because
it would have taken you

00:08:48.660 --> 00:08:51.610
forever to write all
of the scenarios down.

00:08:51.610 --> 00:08:56.450
In fact, there would have been
1,024 scenarios to write down.

00:08:56.450 --> 00:09:00.000
So doing this
exercise for 10 flips

00:09:00.000 --> 00:09:02.484
would have taken
up all of our time.

00:09:02.484 --> 00:09:04.900
But when you think about in a
slightly different way, when

00:09:04.900 --> 00:09:07.260
you just say, look the
probability of getting at least

00:09:07.260 --> 00:09:09.140
1 heads in 10 flips
is the same thing

00:09:09.140 --> 00:09:11.370
as the probably of
not getting all tails.

00:09:11.370 --> 00:09:13.110
And that's 1 minus
the probability

00:09:13.110 --> 00:09:14.710
of getting all tails.

00:09:14.710 --> 00:09:18.867
And this is actually a pretty
easy thing to think about.


Compound probability of independent events

WEBVTT
Kind: captions
Language: en

00:00:00.579 --> 00:00:02.120
Let's think about
the situation where

00:00:02.120 --> 00:00:04.740
we have a completely
fair coin here.

00:00:04.740 --> 00:00:05.520
So let me draw it.

00:00:05.520 --> 00:00:07.190
I'll assume it's a
quarter or something.

00:00:07.190 --> 00:00:07.689
Let's see.

00:00:07.689 --> 00:00:09.700
So this is a quarter.

00:00:09.700 --> 00:00:14.100
Let me draw my best attempt at
a profile of George Washington.

00:00:14.100 --> 00:00:15.270
Well, that'll do.

00:00:15.270 --> 00:00:16.339
So it's a fair coin.

00:00:16.339 --> 00:00:18.130
And we're going to flip
it a bunch of times

00:00:18.130 --> 00:00:20.130
and figure out the
different probabilities.

00:00:20.130 --> 00:00:21.880
So let's start with a
straightforward one.

00:00:21.880 --> 00:00:23.220
Let's just flip it once.

00:00:23.220 --> 00:00:25.160
So with one flip
of the coin, what's

00:00:25.160 --> 00:00:27.150
the probability
of getting heads?

00:00:27.150 --> 00:00:30.320
Well, there's two equally
likely possibilities.

00:00:30.320 --> 00:00:33.490
And the one with heads is one
of those two equally likely

00:00:33.490 --> 00:00:35.960
possibilities, so
there's a 1/2 chance.

00:00:35.960 --> 00:00:37.550
Same thing if we
were to ask what

00:00:37.550 --> 00:00:39.770
is the probability
of getting tails?

00:00:39.770 --> 00:00:43.450
There are two equally likely
possibilities, and one of those

00:00:43.450 --> 00:00:46.260
gives us tails, so 1/2.

00:00:46.260 --> 00:00:48.390
And this is one
thing to realize.

00:00:48.390 --> 00:00:50.250
If you take the
probabilities of heads

00:00:50.250 --> 00:00:52.000
plus the probabilities
of tails, you

00:00:52.000 --> 00:00:54.900
get 1/2 plus 1/2, which is 1.

00:00:54.900 --> 00:00:57.430
And this is generally true.

00:00:57.430 --> 00:01:00.890
The sum of the probabilities
of all of the possible events

00:01:00.890 --> 00:01:03.240
should be equal to 1.

00:01:03.240 --> 00:01:05.710
And that makes sense,
because you're adding up

00:01:05.710 --> 00:01:08.492
all of these fractions,
and the numerator

00:01:08.492 --> 00:01:10.450
will then add up to all
of the possible events.

00:01:10.450 --> 00:01:12.780
The denominator is always
all the possible events.

00:01:12.780 --> 00:01:13.960
So you have all
the possible events

00:01:13.960 --> 00:01:15.334
over all the
possible events when

00:01:15.334 --> 00:01:17.220
you add all of these things up.

00:01:17.220 --> 00:01:19.070
Now let's take it up a notch.

00:01:19.070 --> 00:01:21.477
Let's figure out the
probability of-- I'm

00:01:21.477 --> 00:01:23.060
going to take this
coin, and I'm going

00:01:23.060 --> 00:01:25.270
to flip it twice--
the probability

00:01:25.270 --> 00:01:28.320
of getting heads and then
getting another heads.

00:01:32.570 --> 00:01:34.780
There's two ways
to think about it.

00:01:34.780 --> 00:01:36.410
One way is to just
think about all

00:01:36.410 --> 00:01:37.860
of the different possibilities.

00:01:37.860 --> 00:01:39.860
I could get a head on
the first flip and a head

00:01:39.860 --> 00:01:42.110
on the second flip,
head on the first flip,

00:01:42.110 --> 00:01:43.690
tail on the second flip.

00:01:43.690 --> 00:01:48.130
I could get tails on the first
flip, heads on the second flip.

00:01:48.130 --> 00:01:50.630
Or I could get
tails on both flips.

00:01:50.630 --> 00:01:54.000
So there's four distinct,
equally likely possibilities.

00:01:56.800 --> 00:01:58.990
And one way to think about
is on the first flip,

00:01:58.990 --> 00:02:00.530
I have two possibilities.

00:02:00.530 --> 00:02:02.750
On the second flip, I have
another two possibilities.

00:02:02.750 --> 00:02:04.900
I could have heads or
tails, heads or tails.

00:02:04.900 --> 00:02:06.639
And so I have four
possibilities.

00:02:06.639 --> 00:02:08.930
For each of these possibilities,
for each of these two,

00:02:08.930 --> 00:02:10.699
I have two possibilities here.

00:02:10.699 --> 00:02:14.550
So either way, I have four
equally likely possibilities.

00:02:14.550 --> 00:02:17.300
And how many of those
meet our constraints?

00:02:17.300 --> 00:02:19.120
Well, we have it
right over here,

00:02:19.120 --> 00:02:20.620
this one right
over here-- having

00:02:20.620 --> 00:02:24.450
two heads meets our constraints.

00:02:24.450 --> 00:02:26.960
And there's only one
of those possibilities.

00:02:26.960 --> 00:02:29.220
I've only circled one
of the four scenarios.

00:02:29.220 --> 00:02:31.900
So there's a 1/4 chance
of that happening.

00:02:31.900 --> 00:02:34.260
Another way you could
think about this-- and this

00:02:34.260 --> 00:02:36.900
is because these are
independent events.

00:02:36.900 --> 00:02:38.850
And this is a very
important idea

00:02:38.850 --> 00:02:41.260
to understand in
probability, and we'll also

00:02:41.260 --> 00:02:43.620
study scenarios that
are not independent.

00:02:43.620 --> 00:02:46.570
But these are
independent events.

00:02:46.570 --> 00:02:48.870
What happens in the
first flip in no way

00:02:48.870 --> 00:02:50.717
affects what happens
in the second flip.

00:02:50.717 --> 00:02:53.300
And this is actually one thing
that many people don't realize.

00:02:53.300 --> 00:02:55.030
There's something called
the gambler's fallacy, where

00:02:55.030 --> 00:02:56.650
someone thinks if I
got a bunch of heads

00:02:56.650 --> 00:02:58.020
in a row, then all
of a sudden, it

00:02:58.020 --> 00:03:00.186
becomes more likely on the
next flip to get a tails.

00:03:00.186 --> 00:03:01.210
That is not the case.

00:03:01.210 --> 00:03:03.530
Every flip is an
independent event.

00:03:03.530 --> 00:03:05.600
What happened in the
past in these flips

00:03:05.600 --> 00:03:07.870
does not affect the
probabilities going forward.

00:03:12.320 --> 00:03:14.640
The fact you got a heads
on the first flip in no way

00:03:14.640 --> 00:03:17.060
affects that you got a
heads on the second flip.

00:03:17.060 --> 00:03:18.830
So if you can make
that assumption,

00:03:18.830 --> 00:03:20.705
you could say that the
probability of getting

00:03:20.705 --> 00:03:23.180
heads and heads, or
heads and then heads,

00:03:23.180 --> 00:03:27.590
is going to be the same thing
as the probability of getting

00:03:27.590 --> 00:03:31.860
heads on the first flip
times the probability

00:03:31.860 --> 00:03:34.197
of getting heads
on the second flip.

00:03:34.197 --> 00:03:36.780
And we know the probability of
getting heads on the first flip

00:03:36.780 --> 00:03:39.280
is 1/2 and the probability
of getting heads

00:03:39.280 --> 00:03:41.170
on the second flip is 1/2.

00:03:41.170 --> 00:03:43.240
And so we have 1/2
times 1/2, which

00:03:43.240 --> 00:03:46.560
is equal to 1/4, which is
exactly what we got when

00:03:46.560 --> 00:03:48.610
we tried out all of the
different scenarios, all

00:03:48.610 --> 00:03:51.230
of the equally
likely possibilities.

00:03:51.230 --> 00:03:53.200
Let's take it up another notch.

00:03:53.200 --> 00:03:55.600
Let's figure out
the probability--

00:03:55.600 --> 00:03:57.400
and we've kind of
been ignoring tails,

00:03:57.400 --> 00:03:59.570
so let's pay some
attention to tails.

00:03:59.570 --> 00:04:02.110
The probability of
getting tails and then

00:04:02.110 --> 00:04:06.960
heads and then tails-- so
this exact series of events.

00:04:06.960 --> 00:04:11.080
So I'm not saying in any
order two tails and a head.

00:04:11.080 --> 00:04:13.990
I'm saying this exact order--
the first flip is a tails,

00:04:13.990 --> 00:04:17.690
second flip is a heads, and
then third flip is a tail.

00:04:17.690 --> 00:04:20.510
So once again, these are
all independent events.

00:04:20.510 --> 00:04:22.854
The fact that I got tails
on the first flip in no way

00:04:22.854 --> 00:04:24.270
affects the
probability of getting

00:04:24.270 --> 00:04:25.475
a heads on the second flip.

00:04:25.475 --> 00:04:27.225
And that in no way
affects the probability

00:04:27.225 --> 00:04:29.364
of getting a tails
on the third flip.

00:04:29.364 --> 00:04:31.030
So because these are
independent events,

00:04:31.030 --> 00:04:33.238
we could say that's the same
thing as the probability

00:04:33.238 --> 00:04:35.950
of getting tails on
the first flip times

00:04:35.950 --> 00:04:39.460
the probability of getting
heads on the second flip times

00:04:39.460 --> 00:04:42.530
the probability of getting
tails on the third flip.

00:04:42.530 --> 00:04:45.040
And we know these are
all independent events,

00:04:45.040 --> 00:04:52.300
so this right over here is
1/2 times 1/2 times 1/2.

00:04:52.300 --> 00:04:54.390
1/2 times 1/2 is 1/4.

00:04:54.390 --> 00:05:00.100
1/4 times 1/2 is equal to
1/8, so this is equal to 1/8.

00:05:00.100 --> 00:05:01.080
And we can verify it.

00:05:01.080 --> 00:05:03.690
Let's try out all of the
different scenarios again.

00:05:03.690 --> 00:05:06.930
So you could get
heads, heads, heads.

00:05:06.930 --> 00:05:09.650
You could get
heads, heads, tails.

00:05:09.650 --> 00:05:12.700
You could get
heads, tails, heads.

00:05:12.700 --> 00:05:17.385
You could get
heads, tails, tails.

00:05:17.385 --> 00:05:21.574
You can get tails, heads, heads.

00:05:21.574 --> 00:05:22.990
This is a little
tricky sometimes.

00:05:22.990 --> 00:05:24.865
You want to make sure
you're being exhaustive

00:05:24.865 --> 00:05:27.580
in all of the different
possibilities here.

00:05:27.580 --> 00:05:33.260
You could get
tails, heads, tails.

00:05:33.260 --> 00:05:36.690
You could get
tails, tails, heads.

00:05:36.690 --> 00:05:39.570
Or you could get
tails, tails, tails.

00:05:39.570 --> 00:05:43.120
And what we see here is
that we got exactly eight

00:05:43.120 --> 00:05:45.320
equally likely possibilities.

00:05:45.320 --> 00:05:48.050
We have eight equally
likely possibilities.

00:05:48.050 --> 00:05:51.260
And the tail, heads, tails
is exactly one of them.

00:05:51.260 --> 00:05:53.870
It is this possibility
right over here.

00:05:53.870 --> 00:05:59.060
So it is 1 of 8 equally
likely possibilities.