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I have been playing a game and came up with this question:

There are $n$ different object, and each time you randomly choose one of them.

One success is defined as one of the objects being selected $m$ times. What is the expected time to get one success (accumulating $m$ of any one object)?

In addition, what is the expected rate of success?

Please help...

user103828
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BetaMiao
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  • It doesn't make sense to talk about an expected time if there are no information about the length of time intervals between each selection. –  Feb 09 '15 at 18:54
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    In case you want to calculate the expected number of trials, look up: Binomial Distribution. –  Feb 09 '15 at 18:55
  • See Byron's answer here for an elegant expression for the expected number of trials. – HammyTheGreek Feb 10 '15 at 00:57

1 Answers1

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Here's some thoughts. You are probably interested in the multinomial distribution, since you can accumulate $m$ of any object, and not just $m$ of a single object.

So if $p_k$ is the probability of selecting the $k$th object, and if $X_k$ counts the number of times you have selected object $k$, then you are interested in e.g. $P(X_k=m,X_j<m\text{ for }j\neq k)$ for $k=1,2,\ldots,n$.

Mankind
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