Probability of having outcome of no overlap?

Question

I’m currently having problems understanding how one would calculate the probability (expected value) of having an outcome of no overlap.

1. For instance, how would one calculate the probability of the probability of there being no overlap of birthdays given there is a sample of 100 days, 365 days in a year, and each day has an equal probability of being a birthday of someone.

2. Then taking (1) into consideration, how would I calculate the expected number of distinct birthdays?

Thank you!

Answer 1

For $i=1,2,\dots,365$ let $X_i$ be the random variable whose value is $1$ if at least one person is born on day $i$ and $0$ otherwise, and let $$X=\sum_{i=1}^{365}X_i.$$ Then we want $$E(X)=E\left(\sum_{i=1}^{365}X_i\right)=\sum_{i=1}^{365}E(X_i)=365\left(1-\left(364\over365\right)^{100}\right)\approx87.575518$$