Task 4: Birthday ties

Suppose there are n students in a statistics course group. We compare their birthdays (day and month) and ask how likely it is to have a birthday tie (that is, at least two students with the same birthday). For simplicity, we ignore leap years, i.e assume that there are 365 possibilities for a birthday, all being equally likely (i.e each day having probability 1/365 to be a birthday for a randomly-picked person).

Denoting this probability by p_n  ⁣, it grows with n  ⁣, the number of students in the group. It seems to be quite low for n = 2, but with n bigger than 365, a birthday tie becomes certain, for example, p₃₆₆ = 1.

The question we would like to address is: smallest number n.

What is the smallest number n = n* for which the probability p_n is bigger than 0.5 (i.e it is more likely than not that there is a birthday tie)?

To investigate, perform the following tasks:

• Design and carry out a simulation to evaluate the probability p_n   ⁣of a birthday tie for a given n.
• Experiment with different n  to identify the median value n = n*. Are you surprised by the result?
• Evaluate the MOE of the estimated median probability. To make a confident prediction that this probability is indeed bigger than 0.5, you may need to increase the number of replications.