there are four jars, A through D, and $N$ balls. What is the expected number of jars that are empty if $N=4$. What is the expected number of empty jars if $N \sim Poisson(4)$ The first part I think I got as $\frac{81}{64}$, but I don't know how the second part differs if at all.
2026-04-24 03:28:13.1777001293
Expectation: empty jars
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The second part has the number of balls $N$ distributed according to a Poisson distribution with mean $4$. That means there is $\frac {4^ke^{-4}}{k!}$ chance of having $k$ balls. You just need to do the same analysis you did before for various $k$, multiply by the probability of that value of $k$ and sum them up.