# Probability with birthdays in a specific time range

Out of a group of 71 people, how many of them would have a birthday through July 16th- July 31st? I'm not looking for just one persons probability in the group of having their birthday during that time, more like an overall probability. I have the probability of one person having their birthday during that time, which is 2.12%, but that doesn't tell me about the other 70 people.

• You might ask about the probability that at least one of them had a birthday in that range. Or you could ask about the expected number of them that have a birthday in that range. (The expected number may not be an integer!) Or you could ask about the most likely number of people with a birthday in that range (the "mode"). Do you know the way to calculate any of these? Mar 7 '19 at 7:27
• Also, half a month should be around $4\%$ of the year, not $2\%$. Mar 7 '19 at 7:48
• This range is $\frac {16}{365}$ of all the birthdays so you expect $\frac{16}{365}$ of the people to be in that range. So you expect $71\times \frac {16}{365}$ people to have birthdays in that range. Mar 7 '19 at 21:56

## 1 Answer

Assuming the people's birthdays are independent and uniformly distributed throughout the year, then the number of birthdays can be modeled as a Binomial random variable $$X \sim \text{Binomial}(71,(31-15)/365).$$

The expected value and variance of this random variable are $$E[X] = 71 \cdot \frac{16}{365} \approx 3.11$$ $$\text{Var}(X) = 71 \cdot \frac{16}{365} \cdot \frac{350}{365} \approx 2.98,$$ so we should expect to have the number of birthdays to be in the range $$3.11 \pm 2.98$$ with high probability (68%).

• Simulation suggests $3\pm 2$ at $86.7\%$ Mar 7 '19 at 23:06