# Sock picking without replacement (Probability)

Question:

The chance of picking a red sock out of a drawer of infinite socks is $1\over3$ and the chance of picking a blue sock is $2\over3$

What's the chance that if I pick $20$ socks out of these, $19$ are blue?

Attempt:

I tried to find the probability of $P(\text{Blue} = 19 \text{ & Red} = 1)$ and multiplying it by the number of ways this could happen.

So,

$${P(\text{Blue} = 19 \text{ & Red} = 1)} = {2\over3}^{19} \cdot {1\over3}^1 = 0.00001504$$ Permutations: $\frac{20!}{19!} = 20$

Solution $= 20*0.00001504.$

I know this is wrong because I tried the above procedure with $P(\text{Blue} = 6 \text{ & Red} = 3)$, which intuitively should work out to 1, but did not get the result.

What am I missing here?

• Your calculation seems right to me. Can you explain why P(blue = 6 and red = 3) should be 1? I don't see why that should happen, unless I am missing something. – Abhiram Natarajan Nov 17 '17 at 20:34