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May 2nd, 2018, 06:08 AM   #1
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Find the probability for the event

Hello!

Can someone help me solve this task. My brain ain't functioning anymore and I have no idea where to start



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May 5th, 2018, 04:15 AM   #2
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The "complement event" is that there exist at least one empty car and the "Poincare-Sylvester formula" says that the probability of event A or B (or both) happening is the probability that A happens plus the probability that B happens minus the probability they both happen.
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May 5th, 2018, 08:59 AM   #3
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I think the hint in this case leads to a method much harder than necessary.

I guess what they are trying to make you do is come up with the probability of the union of $k$ cars being empty $k \in 1,2,\dots N-1$

A much much easier approach is to note that in order to maintain at least 1 passenger per car there are $K-1$ "slots" between the $K$ passengers that are to be filled with $N-1$ "walls".

This does assume that passengers are identical which in this problem they essentially are as their probabilities of choosing a car are all identical.

So the problem is to determine how many assignments of walls to slots exist and this is simply

$n = \dbinom{K-1}{N-1}$
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