Question

February 29 occurs only in leap years. Years divisible by 4, but not by 100, are always leap years. Years divisible by 100, but not by 400, are not leap years, but years divisible by 400 are leap years.

a) What probability distribution for birthdays should be used to reflect how often February 29 occurs?

b) Using the probability distribution from part (a), what is the probability that in a group of n people at least two have the same birthday?

Answer 1

(a) We need to find the probability that a birthday comes on February 29. Consider a 400 year period- as leap year cycle repeats every 400 years. Number of leap years in this = 400/4 - 4 + 1 = 97. So,

number of days in this period = 365 * 400 + 97 = 146097

In these days February 29 repeats 97 times. So, probability of birthday on February 29 = 97 / 146097

(b) Required probability = 1 - probability that every birthdays are distinct

= 1 - (probability that every birthdays are distinct with 1 birthday being feb 29) - (probability that every birthdays are distinct with none being on February 29)

= 1 - ((365 * 364 * ... 365 - n + 2)/365ⁿ) n  (400/146097)^n-1 (97/146097) -  ((365 * 364 *  ... 365 - n + 1)/365ⁿ) (400/146097)ⁿ

(In the second term, we are picking distinct days and probability of a birthday falling on each such day is 400/146097 for any day other than feb 29 and 97/146097 for feb29) (n is used since birthday of any of the n person can come on feb 29)

= 1 - ₃₆₅P_n-1 n  (400/146097)^n-1 (97/146097) - ₃₆₅P_n  (400/146097)ⁿ