Warmup Problem for Problem Set 2 (18.440)
1. There are n people in a room. Find the probability that at least two people
have a birthday on the same day.
Hint:
• First observe that this is the same as the complement of the event where
no two people have a birthday on the same day (why do you think the
complement would be easier to work with?).
• Then pick the sample space S to be the days on which each person has a
birthday. For example, "Rebecca and Daniel have a birthday on July 5th,
and Charles has a birthday on June 12th" is an outcome in the sample
space S if n = 3. Notice that we did not say whether Rebecca or Daniel
was born at an earlier time of day, just that their birthdays fall on the
same day: order does not matter for each individual day.
• Then find the size of S.
• Finally, find the number of events in S where no two people have the same
birthday.
Answer: P(at least two people have b-day on same day) = 1- P(no two people
have b-day on same day) = 1 −
365!
(365−n)!
|S|
1
=1−
365!
(365−n)!
365n