Ten balls in three boxes

Topic: Random variables and distributions · more questions on this topic
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Ten balls are thrown independently into boxes 1, 2 and 3, with probabilities 1/4, 1/2 and 1/4.

1. Find the probability that the counts are (2,5,3).

2. Find the probability that box 1 is empty.

Worked answer (click to show)

1. The count vector is multinomial:

\frac{10!}{2!\,5!\,3!}\Bigl(\frac14\Bigr)^2\Bigl(\frac12\Bigr)^5\Bigl(\frac14\Bigr)^3=\frac{315}{4096}\approx 0.0769.

2. Each ball misses box 1 with probability 3/4, independently, so the probability is (3/4)^{10}\approx 0.0563.