I have a population of n unique items and am taking a sample of r. I am sampling with replacement. I would like to calculate the probability of sampling any specific item x times give the sample size and population.
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Let 'S' be an event of occurring that specific item, $P(S)=\frac{1}{n}$
Now, from $r$ samples occurrence of $S$ for $x$ times follow binomial distribution.
Hence,$$P(S=x)=\binom{r}{x}\times \Big(\frac{1}{n}\Big)^x\times\Big(\frac{n-1}{n}\Big)^{r-x}$$
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1$\begingroup$ This of course assumes that each item has an equal probability of selection. $\endgroup$ Apr 19, 2015 at 7:12
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