Matching Birthdays

(Adapted from Statistics: The Art and Science of Learning from by Chris Franklin’s Statistics. Section 5.4, page 229.)

On average, there are 365 days in a given year (excluding leap years). What is the probability that one of your classmates has the same birthday? Since there are 365 days in a year, our intuition tells us that there is probably a small chance that you share a birthday with another student in your class.

Question for students:

Let’s say that there are 25 students total in your class. What is the probability that at least two students share the same birthday?

Working through it:

Keep in mind that the probability of at least one match includes 1 match, 2 matches, 3 matches, etc. Evaluating each probability would take a lot of work. To make it simpler, we can find the probability of at least one match by finding the complement probability of no matches. Note:

P(no matches) + P(at least one match) = 1

P(at least one match) = 1 – P(no matches)

· Suppose the class only has 2 students. The first student’s birthday could be any of the 365 days in a year. So the chance that the second student has a different birthday is 364/365. From our note above, the probability that the two students have the same birthday is,

· Suppose the class has 3 students. The probability that all three have different birthdays is

P(no matches) = P(students 1, 2, and 3 have different birthdays)

= P(students 1 and 2 are different) x P(student 3 is different|students 1 and 2 are different)

The probability that student 3 has a different birthday from students 1 and 2 is 363/365 since there are 363 days left that are different from student 1 and 2.

· We can see that the method follows when we add more students to our classroom. Now consider 25 students in your class. Following the same method, by the time you arrive at the 25^th student, to have a different birthday he/she will have 24 less days to choose from out of 365. So the probability of the 25^th student having a different birthday from students 1 through 24 is 341/365.

P(no matches) = P(students 1 and 2 and 3 and 4… and 25 have different birthdays)

The product of these probabilities equals 0.43. Now, recall that we can easily find the probability of at least one match by using the complement.

P(at least one match) = 1 – P(no matches)

P(at least one match) = 1 – 0.43

P(at least one match) = 0.57

Conclusion:

Wow! This probability is greater than ½! What do you think? Is this a higher chance than you expected? It may help to consider the idea of the complement: with a class of 25 students, there are 300 possible pairs of students who can share the same birthday. And we know that if we have a large number of possible opportunities for something to happen, then it is highly likely that it will happen.

Extension for Teacher:

Have your students go around and interview each of their classmates to find everyone’s birthday. Once everyone is finished, ask the class if they found any students who had matching birthdays. Or, announce the concept of the activity to the class. Then starting on one side of the room, ask each student to state their birthday. See how far across the room you can get until you (possibly) find a match. They may be surprised! Take note of your students’ predictions and reactions.

Note: This works best when you have a class of at least 23 students. For a class of at least 23 students, the probability that there is at least one match is greater than ½.

References

Agresti, Alan and Franklin, Christine A. (2007). Statistics: The art and science of learning. Upper Saddle River, NJ. Pearson Prentice Hall.