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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.

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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: $$ \Bbb E[X] = 71 \cdot p = 1.167, \quad Var(X) = 71 \cdot p \cdot (1-p) = 1.148 $$

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