Question: Use Poisson approximations to investigate the following types of coincidences. The usual assumptions of the birthday problem apply, such as that there are 365 days in a year, with all days equally likely.

(a) How many people are needed to have a 50% chance that at least one of them has the same birthday as you?

(b) How many people are needed to have a 50% chance that there are two people who not only were born on the same day, but also were born at the same hour (e.g., two people born between 2 pm and 3 pm are considered to have been born at the same hour).

(c) Considering that only 1/24 of pairs of people born on the same day were born at the same hour, why isn't the answer to (b) approximately 24 · 23? Explain this intuitively, and give a simple approximation for the factor by which the number of people needed to obtain probability p of a birthday match needs to be scaled up to obtain probability p of a birthday-birth hour match.

(d) With 100 people, there is a 64% chance that there are 3 with the same birthday (according to R, using pbirth day(100,classes=365,coincident=3) to compute it). Provide two different Poisson approximations for this value, one based on creating an indicator r.v. for each triplet of people, and the other based on creating an indicator r.v. for each day of the year. Which is more accurate?