How many different birthdays
Thirty people have independent birthdays, each uniform over the 365 days of the year.
Find the expected number of different days that are someone's birthday.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Do not count people. Count days: for each day, use an indicator that is 1 if at least one of the thirty was born on it.
The probability that nobody was born on a given day is .
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The number of different birthdays is a sum over the 365 days of an indicator “this day is someone's birthday”. Expectation is linear even though these indicators are dependent, so we only need the probability for one day.
Step 1. For day , let if at least one person was born that day. The number of different birthdays is .
Step 2. A given person misses day with probability , and the thirty are independent, so
Step 3. By linearity,
A check. Thirty people have on average about 1.16 “lost” birthdays: that is the expected number of coincidences, close to pairs sharing a birthday. Yet the probability that at least two people share a birthday is about 71%. Coincidences are likely, but there are usually only one or two of them.