Exactly two balls in one box

Question

Eight balls are thrown independently into 4 boxes, each ball landing in a uniformly random box. What is the probability that box 1 contains exactly 2 of the balls?

Accepted Answer

Track each ball as either "lands in box 1" (probability \frac{1}{4}) or "lands elsewhere" (probability \frac{3}{4}). The 8 balls are independent, so the number landing in box 1 follows a binomial distribution with 8 trials and success probability \frac{1}{4}. For exactly 2 of the 8 balls in box 1: - Choose which 2 balls land there: \binom{8}{2} = 28 ways. - Those 2 land in box 1: probability \left(\frac{1}{4}\right)^2. - The other 6 land elsewhere: probability \left(\frac{3}{4}\right)^6. P = \binom{8}{2}\left(\frac14\right)^2\left(\frac34\right)^6 = 28 \cdot \frac{1}{16}\cdot\frac{729}{4096} …

Exactly two balls in one box

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