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A random function $rand()$ return a integer between $1$ and $k$ with the probability $\frac{1}{k}$. After $n$ times we obtain a sequence $\{b_i\}_{i=1}^n$, where $1\leq b_i\leq k$. Set $\mathbb{M}=\{b_1\}\cup\{b_2\}\cdots \cup\{b_n\}$, $\mathbb{N}=\{b_1\}\cup\{b_2\}\cdots \cup\{b_{n-1}\}$,

I want to calclute the probability satisfy the following condition:

$\mathbb{M}=\{1, 2\cdots, k\}$ and $\mathbb{N}\neq\{1, 2\cdots, k\}$.

Denoting the probality with $P(n)$. If $1\leq n< k$, then $P(n)=0$. $P(n)=?$, where $n\geq k$.

Yuansheng liu
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2 Answers2

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The probability that $j$ of the $k$ numbers is excluded is $(1-j/k)^n$ and there are $\binom{k}{j}$ ways to choose those $j$ to be excluded. The generalized inclusion-exclusion principle says that the probability that exactly one of the $k$ numbers is excluded after $n-1$ trials is $$ \sum_{j=1}^k(-1)^{j-1}j\binom{k}{j}\left(1-\frac{j}{k}\right)^{n-1} =k\sum_{j=1}^k(-1)^{j-1}\binom{k-1}{j-1}\left(1-\frac{j}{k}\right)^{n-1} $$ Then the probability to pick the right one to finish on trial $n$ is $1/k$. Thus, the probability that you've just finished on trial $n$ is $$ \sum_{j=1}^k(-1)^{j-1}\binom{k-1}{j-1}\left(1-\frac{j}{k}\right)^{n-1} $$

Another method

In this answer, it is shown that the probability of still having at least one of the $k$ numbers excluded is $$ \sum_{j=1}^k(-1)^{j-1}\binom{k}{j}\left(1-\frac{j}{k}\right)^n $$ Subtracting from $1$, we get that the probability of getting all $k$ numbers by trial $n$ is $$ \sum_{j=0}^k(-1)^j\binom{k}{j}\left(1-\frac{j}{k}\right)^n $$ To get the probability that we get all $k$ numbers on trial $n$, take the difference between this for $n$ and $n-1$: $$ \begin{align} &\sum_{j=0}^k(-1)^j\binom{k}{j}\left[\left(1-\frac{j}{k}\right)^n-\left(1-\frac{j}{k}\right)^{n-1}\right]\\ &=\sum_{j=0}^k(-1)^{j-1}\binom{k}{j}\frac{j}{k}\left(1-\frac{j}{k}\right)^{n-1}\\ &=\sum_{j=1}^k(-1)^{j-1}\binom{k-1}{j-1}\left(1-\frac{j}{k}\right)^{n-1} \end{align} $$

Community
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robjohn
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Hint: to succeed, $\Bbb N$ has to be missing exactly one element and $b_n$ has to be that element.

Ross Millikan
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  • Tks very mach. I think this hint so public knowlegde. For express the problem more clearly, I discribe it as above. – Yuansheng liu Apr 21 '13 at 00:53
  • @Yuanshengliu: many of our posters would not have seen that, so it would have been helpful for them. Do you have any thoughts on calculating the probability that $|\Bbb M|=k-1$? I have a recurrence approach for specific $k$, but nothing to offer for general $k$. – Ross Millikan Apr 21 '13 at 02:36