Problem: There are 37 students in a classroom. If it was paying even money, would you bet that at least 4 of them would have their birthday in the same month?

Source: Adapted from a puzzle on Thoughts of a Rambler.

Solution:

Firstly, assume the worst case (from your betting perspective) where all the students have birthdays in different months. This is your worst case scenario because it reduces the changes of one month having a cluster of4 or more students. But there are only 12 months in the year so the most evenly distributed assignment of students to birthday months is 3 in each plus 4 in one month, given 3x11 + 4 = 37. In other words, there is no possible distribution of 37 students to 12 months which does not produce at least 1 month which has 4 or more students assigned to it. Given that you cannot lose on this bet - so bet the house!!! Remember - it's not gambling if you know the odds are in your favour!

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