Question

A department has 20 faculty. Assume that, for a fixed faculty (e.g. Sam) and a fixed month (e.g. March) the probability of Sam’s birthday in that month is 1/12 (i.e. any of the months have equal probability) of containing Sam’s birthday. Further assume birthday’s for different faculty are independent following the above characteristics (thus for example probability Brad and Sam both have birthday in March = (1/12) × (1/12) etc).

The HR manager says there can be a party at the end of the month if there is at least one birthday in that month (so each month has at most one party for all the faculty who have birthdays in that month). Let Y = number of parties in the entire year. Find E(Y).

Answer 1

The expected number of birthday parties in a department with 20 faculty members in a year, given that each month has an equal chance to contain a birthday, is found to be approximately 11.46.

Step-by-step explanation:

To find the expected number of parties, E(Y), which a department of 20 faculty members with birthdays uniformly distributed across the year might celebrate, we can use the concept of the probability of at least one birthday in a month. The probability that no faculty member has a birthday in a given month is (11/12)^20. Consequently, the probability of at least one birthday in a month is 1 - (11/12)^20. Since each month has the same probability of containing at least one birthday and there are 12 months in a year, the expected value E(Y) is simply 12 × (1 - (11/12)^20).

To perform the calculation, calculate the probability of no birthdays in one month, subtract from one, and then multiply by the number of months:

1. Calculate the probability of no birthdays in one month: (11/12)^20.
2. Subtract this from 1 to get the probability of at least one birthday: 1 - (11/12)^20.
3. Multiply by 12 to get the expected number of parties in a year: 12 × (1 - (11/12)^20).

Performing the calculations gives us, E(Y) ≈ 12 × (1 - (11/12)^20) ≈ 11.46, which is the expected number of birthday parties in a year.