Illinois State University Mathematics Department

MAT 312: Probability and Statistics for Middle School Teachers Dr. Roger Day (day@math.ilstu.edu) |

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 onthePigeon Hole Principle.

ThePigeon Hole PrincipleIf there are

ncategories by which to classify objects, then no more thann+1objects 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: