A shared birth month
Four people each have a birth month chosen independently and uniformly among the months.
What is the probability that at least two of them were born in the same month?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Compute the probability of the complement: all four months are different.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. “At least two share” has many cases; “all different” has one, and it can be counted by choosing months one person at a time.
Step 1: complement. The number of ways to give four people four different months, in order, is . The total number of assignments is . So
Step 2.
Comment. With only people and months the chance is already above . The reason is that there are pairs, each matching with probability ; the crude estimate overcounts a little because several pairs can match at once.