Birthdays

What is probability that at least two people in Astr505, or two people in the astronomy department, have the same birthday? (We ignore the year of birth, and look only at the day of the year of birth.)

For one person, she may be born on any day of the year and so P1 = 365 / 365 = 1.

For a second person, he may be born on all but one day of the year and so P2 = 364 / 365.

For a third person, she may be born on all but two days of year and so P3 = 363 / 365.

For an nth person, Pn = (365 - n + 1) / 365.

The probability P of different birthdays for all n people is the product of the individual probabilities.

The probability P(n) that at least two of the n people were born on the same day of the year is simply the conjugate of P.

We can easily calculate a few cases, to find the region around P(n) = 50%.

Remember that sometimes it is much easier to calculate the conjugate of the case that you want than to consider every case which meets your criteria.