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I have $N$ bins labeled "1" through "N". I place $k$ balls in bin "1", then successively move balls from bin "1" to bin "2", bin "2" to bin "3", and so on to bin "N". However, while moving any particular ball from one bin to another, I discard the ball with probability $(1 - p)$.

What is the probability distribution for the number of balls in bin "N"?

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The probability that a given ball survives a single move is $p$. Thus the probability $p^\ast$ that it survives the $N-1$ moves required to end up in bin $N$ is $p^{N-1}$.

So the distribution in bin $N$ is binomial, probability of "success" $p^\ast$. The probability we end up with $i$ balls in bin $N$ is $$\binom{k}{i}(p^\ast)^i (1-p^\ast)^{k-i}.$$

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