Question

each of the 20 balls is tossed independently and at random into one of the 5 bins. let p be the probability that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. let q be the probability that every bin ends up with 4 balls. what is p q ?

Answer 1

if p is the probability that some bin ends up with 3 balls and q is the probability that every bin ends up with 4 balls. pq is 16.

First, let us label the bins with 1,2,3,4,5.

Applying multinomial distribution with parameters  n=20  and  p1=p2=p3=p4=p5=15  we find that probability that bin1 ends up with 3, bin2 with 5 and bin3, bin4 and bin5 with 4 balls equals:

5−2020!3!5!4!4!4!

But of course, there are more possibilities for the same division  (3,5,4,4,4)  and to get the probability that one of the bins contains 3, another 5, et cetera we must multiply with the number of quintuples that has one 3, one 5, and three 4's. This leads to the following:

p=20×5−2020!3!5!4!4!4!

In a similar way we find:

q=1×5−2020!4!4!4!4!4!

So:

pq=20×4!4!4!4!4!3!5!4!4!4!=20×45=16

thus, pq = 16.

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