If we roll 4 dices (fair), what is the probability of "sum of subset" being 5. e.g. 1432,1121, 2344, 2354 have a subset sum of 5. Can you illustrate how to calculate this.



probability of subset sum after rolling dice 4 times
The most straightforward way is also the most tedious: list all die rolls (1296 of them,
I assume), and count which ones have a subset sum of 5. You can save on some work by
looking at all rolls which have at least one 5 in them, and then all rolls which have no 
5 but at least one 1 and one 4 or one 2 and one 3, and then enumerate the remaining cases.


A possibly quicker but still involved approach involves calculating the probabilities
that a sum appear on the kth roll but not the (k-1)th roll. Then one gets 1/6
of a probability of a sum on the first roll and 1/6 + 1/4 = 5/12 of a probability of
such a sum on the first or the second roll. The calculation for the third roll assumes
that no subset sum has occurred on the scone roll, and breaks down into 21 cases, starting
with 6 6 and going down to 1 1, if you like, but many of the cases can be treated the
same way, so that there are fewer types of cases to consider.


You can also try looking at rolls which miss having a subset sum of 5. Taking all even
die gets you a lower bound of 1/16 for not having a subset sum of 5, while looking
at all combinations of 3, 4, and 6, and removing the even combinations gives you (81-16)
more rolls which do not have a subset sum of 5, giving a lower bound of (1/8 -1/81) for
missing a subset sum of 5. There are still the extra cases to handle.


I do not see a simple characterization that allows a quick calculation of the
probability.


Gerhard "Ask Me About System Design" Paseman, 2010.05.19


