Let's say that I have a group of 71 people and I'm trying to find the expected value of how many of their birthdays fall within a specific 6 day time interval (June 12-17).
Would this type of problem work as a binomial random variable? If so, what formula would I use to calculate the expected value? Could I then calculate the variance as well? I know that the formula for the expected value is $P(x) \cdot x$, where $P(x)$ is the chance of success and $x$ is the number of trials.
But I am confused as to whether the 71 people is the number of "trials" and what would the chance of success be? Would it change with each person? Thanks in advance.
you could define random variables $X_i$ as $$ X_i = \begin{cases}1, & \text{person $i$ has birthday within those 6 days} \\ 0, & \text{otherwise} \end{cases}. $$
Then clearly $X_i$ has a Bernoulli distribution with $$ \Bbb P (X_i = 1) = \frac{6}{365} =:p. $$
Now if you assume that all $X_i$ are independent (e.g. there are no twins etc in your group), then $X = \sum_{i = 1}^{71}X_i$ has a Binomial distribution: $$ X \sim \text{Bin}(71, p). $$
The expected value and variance can then be easily computed: $$ \Bbb E[X] = 71 \cdot p = 1.167, \quad Var(X) = 71 \cdot p \cdot (1-p) = 1.148 $$