Illinois State University Mathematics Department

MAT 312: Probability and Statistics for Middle School Teachers

Dr. Roger Day (

The Birth Month Problem

We can argue that for this situation, the probability will be 1 provided there are more than 12 people in the room. This conclusion is based on the Pigeon Hole Principle.

The Pigeon Hole Principle

If there are n categories by which to classify objects, then no more than n+1 objects are required to assure that at least two of the objects share a category.

Suppose a class has more that 12 students. Under a worst-case scenario, the first 12 students could each name a different birth month. Upon questioning, however, the 13th student must name a birth month already stated by one of the first 12 students.

For the Birth Month Problem, a graphical representation comparing the number of people in the room and the associated probability of at least one match of a birth month is shown below.

We see that the range of values appropriate for expressing the probability is from 0 to 1 inclusive. This will always be the case for probability situations: