Odds that 5 persons share the same last name given a group of n people What are the odds that 5 individuals share the same last name, say Miller, in a group of 50 assuming the associated probability of 'Miller' in a population being 2%.
How do I calculate it? Thought about using binomial probability density function, but I duno whether this is actually correct.
 A: Using a binomial model is appropriate if the 50 people are randomly sampled from a population in which 2% of the people are named Miller. (If this is a gathering to celebrate the 40th wedding anniversary of Tim and Mary Miller, then a binomial model would not be appropriate.)
Let $X$ be the number of Millers in the sample of 50. Then $X \sim \mathsf{Binom}(n=50,p=0.02).$ Then if you seek $P(X = 5) = 0.00273,$ you can compute the result in R, where dbinom is a binomial PDF, as follows:
 dbinom(5,50,.02) 
 # 0.002731525. 

Using the binomial PDF, this is
$$P(X = 5) = {50\choose 5}(.02)^5(.98)^{45}.$$
If you mean 'at least five', then you need to compute
$P(X \ge 5) = 1 - P(X \le 4)$ and I will leave that computation from the binomial PDF to you. From R, the answer is as follows:
1 - pbinom(4, 50, .02)
[1] 0.003209742

A normal approximation does not work well here.
mu = 50*.02;  sg = sqrt(50*.02*.98)
1 - pnorm(4.5, mu, sg)
[1] 0.000203476     # with continuity correction
1 - pnorm(4, mu, sg)
[1] 0.001220917     # without


