Probability of choosing $k$ distinct objects when choosing $n$ objects from $n$ objects with replacement.

Question

Assume we have $n$ objects. And we are selecting $n$ objects from these $n$ objects with replacement, where the probability of choosing any object is $\frac{1}{n}$.

For each $k$ from $1$ to $n$, what is the probability that we chose $k$ distinct objects.

For $k = 1$, this is clearly $(\frac{1}{n})^n$.

For $k = 2$, this is $(\frac{1}{n})^{n-1}(1-\frac{1}{n})\binom{n}{1}$.

But I can't seem to generalize this for $2 < k \leq n$. And I see that there is a similar question regarding expectation given here. But I am trying to calculate individual probabilities. Expected number of unique items when drawing with replacement

Answer 1

The Stirling Number of the Second Kind ${n\brace k}$ counts the number of ways to partition an $n$ element set into $k$ non-labeled non-empty subsets.

The falling factorial $n\frac{k}{~}$ is the number of ways to select an ordered sequence of $k$ elements out of $n$ with no repeats.

So, to count the ways to have $k$ distinct objects selected in your $n$ selections, first select a way to partition the sequence of selections (first time you picked, second time you picked, etc...) into $k$ non-labeled non-empty subsets (e.g. the first subset being the first select, third selection, fifth selection) so that each time in the selection process all of those times in the same part in the partition will have had the result of their selection be the same. Then, choose what the selection was for each of your groups of selections.

$$\Pr(X=k)=\frac{n\frac{k}{~}{n\brace k}}{n^n}$$

Answer 2

Alternative approach:

The Math in the answer of JMoravitz can be derived analytically, using Inclusion-Excusion. See this article for an introduction to Inclusion-Exclusion. Then, see this answer for an explanation of and justification for the Inclusion-Exclusion formula.

For any set $E$ with a finite number of elements, let $|E|$ denote the number of elements in the set $E$.

Assume that $N \in \Bbb{Z_{\geq 2}}$ and that $K \in \{1,2,\cdots,(N-1)\}.$ Here, it is being assumed that $N$ and $K$ are fixed positive integers.

Let $A$ denote the set of all possible ordered $N$-tuples $\left(a_1, a_2, \cdots, a_N\right)$, where each component $a_i$ is an element in $\{1,2,\cdots,N\}$.

Then, each element in $A$ represents a distinct way that $N$ items can be selected from $\{1,2,\cdots,N\}$, sampling with replacement, where the order of the selection is deemed important.

Then $|A| = N^N.$

Let $B$ denote the subset of $A$, where each ordered $N$-tuple $\left(a_1, a_2, \cdots, a_N\right) \in B$ satisfies the following constraints:

• Each component $a_i$ is an element in $\{1,2,\cdots,K\}$.

• For each element $m$ in the set $\{1,2,\cdots,K\}$ at least one of the components $a_1, a_2, \cdots, a_N$ is equal to $m$.

Then, the desired computation of the probability is

$$ \frac{\binom{N}{K} \times |B|}{|A|} = \frac{\binom{N}{K} \times |B|}{N^N}. \tag1 $$

When examining whether order of selection is to be regarded as important, the numerator and denominator in (1) above must be computed in a consistent manner. Further, it is very convenient to regard order of selection as important, when (for example) enumerating $A$. This convenience drives my strategy.

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In (1) above, the factor of $\binom{N}{K}$ in the numerator reflects that any $K$ items from $\{1,2,\cdots,N\}$ could be chosen to be the $K$ items that will be selected. Note that this approach takes advantage of the fact that $B$ represents that each of the items in $\{1,2,\cdots,K\}$ will be selected at least once.

Therefore, you have $\binom{N}{K}$ mutually exclusive subsets of ordered $N$-tuples, where each subset represents that $K$ specific elements from $\{1,2,\cdots,N\}$ will be selected.

So, based on (1) above, the problem has been reduced to computing $|B|$.

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Let $S$ denote the subset of $A$, where each ordered $N$-tuple $\left(a_1, a_2, \cdots, a_N\right) \in S$ satisfies the following constraint:

• Each component $a_i$ is an element in $\{1,2,\cdots,K\}$.

Notice that the set $S$ is a superset to the set $B$, and that the set $S$ will (also) include ordered $N$-tuples whose components do not span $\{1,2,\cdots,K\}$.

For $j \in \{1,2,\cdots,K\}$ let $S_j$ denote the subset of ordered $N$-tuples from $S$ that each satisfy the following constraint:

• None of the components of the ordered $N$-tuple is equal to $j$.

Then

$$|B| = |S| - |S_1 \cup S_2 \cup \cdots \cup S_K|. \tag2 $$

Let $T_0$ denote $|S|$.

Let $T_1$ denote $~\displaystyle \sum_{1 \leq i_1 \leq K} |S_{i_1}|.$
Thus, $T_1$ denotes the summation of $~\displaystyle \binom{K}{1}$ terms.

Let $T_2$ denote $~\displaystyle \sum_{1 \leq i_1 < i_2 \leq K} |S_{i_1} \cap S_{i_2}|.$
Thus, $T_2$ denotes the summation of $~\displaystyle \binom{K}{2}$ terms.

Similarly, for $r \in \{3,4,\cdots,(K-1)\}$
let $T_r$ denote $~\displaystyle \sum_{1 \leq i_1 < i_2 < \cdots < i_r \leq K} |S_{i_1} \cap S_{i_2} \cap \cdots \cap S_{i_r}|.$
Thus, $T_r$ denotes the summation of $~\displaystyle \binom{K}{r}$ terms.

Then, in accordance with Inclusion-Exclusion theory,

$$|B| = \sum_{r=0}^{K-1} (-1)^r T_r.$$

So, the problem is reduced to computing each of
$T_0, T_1, \cdots, T_{K-1}.$

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$\underline{\text{Computation of} ~T_0}$

There are $K$ choices for each component of the ordered $N$-tuple in $S$. Therefore,

$$T_0 = |S| = K^N.$$

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$\underline{\text{Computation of} ~T_1}$

Similar to the analysis in the previous section, when enumerating $S_1$, there are $(K-1)$ choices for each component of the ordered $N$-tuple in $S_1$. Therefore,

$\displaystyle |S_1| = (K-1)^N.$

Further, by symmetry, $|S_1| = |S_2| = \cdots = |S_K|.$

Therefore,

$$T_1 = \binom{K}{1} \left(K-1\right)^N.$$

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$\underline{\text{Computation of} ~T_2}$

Similar to the analysis in the previous section, when enumerating $\left(S_1 \cap S_2\right)$, there are $(K-2)$ choices for each component of the ordered $N$-tuple in $\left(S_1 \cap S_2\right)$. Therefore,

$\displaystyle |S_1 \cap S_2| = (K-2)^N.$

Further, by symmetry, for each $1 \leq i_1 < i_2 \leq K,$ you have that $|S_{i_1} \cap S_{i_2}| = |S_1 \cap S_2|.$

Therefore,

$$T_2 = \binom{K}{2} \left(K-2\right)^N.$$

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$\underline{\text{Computation of} ~T_r ~: 3 \leq r \leq (K-1)}$

Similar to the analysis in the previous section, when enumerating $\left(S_1 \cap S_2 \cap \cdots \cap S_r\right)$, there are $(K-r)$ choices for each component of the ordered $N$-tuple in $\left(S_1 \cap S_2 \cap \cdots \cap S_r\right)$. Therefore,

$\displaystyle |S_1 \cap S_2 \cap \cdots \cap S_r| = (K-r)^N.$

Further, by symmetry, for each $1 \leq i_1 < i_2 < \cdots < i_r \leq K,$ you have that $|S_{i_1} \cap S_{i_2} \cap \cdots \cap S_{i_r}| = |S_1 \cap S_2 \cap \cdots \cap S_r|.$

Therefore,

$$T_r = \binom{K}{r} \left(K-r\right)^N.$$

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Final computation:

$$|B| = \sum_{r=0}^{K-1} (-1)^r \times T_r = \sum_{r=0}^{K-1} \left[(-1)^r \times \binom{K}{r} \left(K-r\right)^N\right]. \tag3 $$

Combining (3) and (1), the desired computation of the probability is

$$ \frac{\binom{N}{K} \times |B|}{N^N}, $$

where $|B|$ is computed in (3) above.