# Sampling with replacement - Expected number of duplicates, triplicates, …, n-tuples

I would like to create an estimate for the expected number of types of gene repeats when drawing from a set of genes where each gene is unique and has equal probability.

i.e.
I have a genome of size N
I have a sample of size m genes
Each gene has a probability 1/N
I define a duplicate as seeing the same gene sampled twice, triplicate thrice, etc
So, from a subset {2,2,3,4,8,9,9,9,9,1,1} taken from some set, you see 2 duplicates, 0 triplicates, and 1 quadruplicate

If I sample with replacement, how many duplicates would I expect to see? How can I expand this to create a distribution of expected pairs, triples, quadruples,..., ntuples

Thank you!

If you have $N$ equally probable genes and sample $m$ times with replacement then the probability of seeing a particular gene exactly $k$ times is (from the binomial distribution): ${m \choose k}\frac{(N-1)^{m-k}}{N^m}$ and so (by linearity of expectation) the expected number of genes appearing exactly $k$ times is $${m \choose k}\frac{(N-1)^{m-k}}{N^{m-1}}.$$

In your example $m=11$. If $N=9$ then the expected numbers for different $k$ are about

 k  Expected number
0  2.463569
1  3.387408
2  2.117130
3  0.793924
4  0.198481
5  0.034734
6  0.004342
7  0.000388
8  0.000024
9  0.0000010
10  0.000000025
11  0.00000000029