Problem 8. Let X be the number of distinct birthdays among four persons selected
ID: 2923682 • Letter: P
Question
Problem 8. Let X be the number of distinct birthdays among four persons selected uniformly at random. Assume that all years have 365 days and birthdays are randomly distributed throughout the year. Describe the random variable X.
Hint: Rng(X) = {1, 2, 3, 4}. Here (not all possibilities, just some examples) is what I mean by distinct birthdays:
"1 distinct birthday": Alice, Bob, Raul, and Yi-Fei were all born on January 1st;
"2 distinct birthdays": Alice and Bob were born on Jan 1st, but Raul and Yi-Fei were born on July 23rd;
"2 distinct birthdays": Alice was born on Jan 1st, but Bob, Raul, and Yi-Fei were born on July 23rd;
"3 distinct birthdays": Alice and Bob were born on Jan 1st, Raul was born on July 3rd, and Yi-Fei was born on Dec 4th
"4 distinct birthdays": All 4 were born on different days.
It is simpler to think of this as choosing a sequence of k birthdays, and then figuring out the number of patterns of shared birthdays. You should consider the probability of all possible patterns for each of these, e.g., 1 birthday is {A, A, A, A} and 2 distinct birthdays is either {A, B, B, B} or {A, A, B, B}, and so on. Be careful of overcounting!
Explanation / Answer
A person can be born on any of the 365 days of a year. Also, there are 365^4 total possibilities because there are four people in total: -
1 distinct birthday: 365/365^4= 0.000000021
2 distinct birthday: 4C2*365*364/365^4=0.0000449
2 distinct birthday: 4C2*365*364*363/365^4=0.01630
3 distinct birthdays: 4C3*365*364*363/365^4=0.01630
4 distinct birthdays: (365 * 364 * 363 * 362) / 365^4=0.983