problem 23

3.2.7 problem 23

Let $I_{i}$ be the indicator of the $i$ -th person voting for Kodos. Then, $P (I_{i} = 1) = p_{1} p_{2} p_{3} .$ Since the voters make their decisions independently, we have $n$ independent Bernoulli trials, which is precisely the story for a Binomial distribution.

Thus,

P (X = k) = (\frac{n}{k}) {(p_{1} p_{2} p_{3})}_{1}^{k} {(1 - p_{1} p_{2} p_{3})}_{1}^{n - k}

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