You walk into a room with 7 other people in it. What is the probability that one or more of them has a birthday in the same month as yours?

Question

You walk into a room with 7 other people in it. What is the probability that one or more of them has a birthday in the same month as yours?

Answer 1

This is an example of a classic probability problem that requires you to know that a probability of 1 is certainty and that the probability of event E not happening is 1 – probability of event E happening.

The probability of you walking into a room and the one person already in the room having a different birth month is 11/12 (assuming birthdays are equally likely in each of 12 months).  The probability of the birthdays being in the same month is 1/12 (or, 1 – 11/12).

 

Now, the probability of you walking into a room with two people already in it and neither of them having the same birth month as you is (11/12)(11/12), so the probability that one or both of them shares your birthday month is (1-(11/12)(11/12)),

 . . .

 

The probability of walking into a room with 7 other people in it and one or moreof them has a birthday in the same month as yours is:

        1 – probability that none of them shares your birthday month

        1 – (11/12)*(11/12)*(11/12)*(11/12)*(11/12)*(11/12)*(11/12)

        0.456149  (or 46%)         (when pasted into MS Excel)

 

    Note: this probability increases so rapidly that the probability is more than 90% when the room has 27 people.