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Say I have T balls, B of them black and T-B of them white.

I have N bins, each with a maximum capacity of T/N (that is, when the balls are all placed into the bins, all bins have the same number of balls, so assume T is always exactly divisible by N).

The balls are randomly assigned to the bins.

What is the probability that each of the bins has at least one black ball?

I figured out the probability that a given bin has a black ball, but I'm quite stuck at how to proceed - it doesn't seem that knowing the probability for a given bin helps me.

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Consider the point of view of a black ball.

It can choose any of the $N$ bins. A bin does not have a black ball if it is not chosen by any of them, and probability that each of the bins does not have a black ball is the intersection $P(A_1\cap A_2\cap \dots A_N )$,

where $A_i$ is the event that bin $i$ does not have a black ball.

Since all $A_i$s are independent(balls choose bins independently randomly uniformly) this probability is the product of $P(A_i)'s$ which are each equal to $$1 - \Big(\frac{N-1}{N}\Big)^B$$ (bin not chosen by a particular black ball, do that for $B$ balls which gives to the power $B$, it's complement gives 'at last one black ball' case. Notice that this probability is independent of number of white balls)

Lastly, you should also subtract the probability of cases in which a bin is chosen by more than $\frac T{N}$ balls, as that is not possible.

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  • $\begingroup$ Since the balls exactly fill the bins to capacity and all bins end up with the same number of balls, this calculation doesn't pertain to the case at hand -- you've shifted the entire work into the last sentence, subtracting out the inadmissible cases, i.e. almost all cases. $\endgroup$ – joriki Jun 7 '18 at 22:24

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