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Introduction: In 1939, a mathematician named Richard von Mises first proposed what we know today as the birthday problem.  He wondered, "How many people must be in a room before the probability that some share a birthday, ignoring the year and ignoring leap days, becomes at least 50 percent?"  We believe that this is one of the most explored probability problems in classrooms today.  This lesson is based on Lawrence Lesserís article that describes the set-up of the spreadsheet simulation and Cindia Stewartís lesson that seeks to answer the Birthday Problem using three different methods.

I. Probability Topic(s): Sample Size, Law of Large Numbers, Complementary Probabilities, Independence of Events  

A. NCTM Standards addressed and link to NCTM: Algebra, Data Analysis & Probability, Problem Solving, Reasoning & Proof, Communication, Connections, and Representations.  http://standards.nctm.org/protoFINAL/chapter7/introduction.html

B. Related Connections: Algebra I (point plotting & graphing), Algebra II (composition of functions), Precalculus (graphical analysis), Calculus (derivatives and Taylor Series)  

 

II. Assumed prior experience with: Basic Probability

 

III. Rationale: To find a solution to a famous Probability problem while using three distinct mathematical methods.

 

IV. Learning Objectives:

1. Students will make predictions and analyze them.

2. Students will answer the birthday problem using three different methods.

3. Students will use the solutions to the birthday problem to answer related questions.

 

V. Materials & Technology Needed:

1. Answer sheet for each student.  This is designed for students to record all of their predictions throughout the lesson in one place.  They can also record their reflections on the lesson on the same sheet

2. Spreadsheet application (Excel), Birthday Problem Spreadsheet Program. We have provided tips for you on using spreadsheet simulations.

3. Graphing calculator (optional)

VI. Procedure:

1. Pose the Birthday Problem in this form: How many people must be in a room before the probability that some share a birthday becomes at least 50 percent?

2. Have students predict an answer to the problem and write it down on their answer sheet.  Do not yet answer any questions they ask to try to clarify the problem (e.g. Does the birth year matter?  What about February 29?)

3. Analyze the conditions: Does a personís birth year matter?  How many days are in a year?  Is a personís birthday equally likely to be any day of the year?  Does the date of one personís birthday affect the date of another personís birthday?  If two people were both born on May 12, is that a match?  If nobody else in the class has the same birthday as me (August 2), does that mean that there are no matches in the classroom?

4. Allow the students to re-evaluate their predictions and make new one if they wish to.

METHOD 1 - SIMULATION

5. Make a list of the name and birthday of everyone in the room.  (This can be done before class to save time.)  Everyone should be able to see the list.  Are there any matches?

6. If there is a match, ask, "Does that mean that there will always be a match in a group of people this large?  Is the probability of a match high with this number of people?"

If there is not a match, ask, "Does that mean that there will never be a match in a group of people this large?  Is the probability of a match low with this number of people?"

7. Some students might have trouble realizing that one trial of an experiment is not sufficient for predicting probabilities.  This could allow for a discussion regarding sample size, the number of trials in an experiment, and the Law of Large Numbers.

8. Open up the spreadsheet program.  Explain to the students how it is doing the same thing the class did in the previous step.  Example: 102 ŗ 102nd day of the year ŗ this personís birthday is April 12.

9. Ask, "Why does the program use 50 trials?"

10. Run the program using the number of people in the classroom for the value of k.  What does the simulation suggest is the probability for a birthday match for that value of k people?

11. Run the program with several other values of k, some low and some high.  Make sure to include k = 23.  If the students have access to the computers, have each student (or group of students) run the program for different values of k.  If there is only one computer, then the teacher can run several experiments with different values of k.

12. What does the program indicate is the answer to the problem?  Discuss this result and the studentsí predictions.  Did anybodyís prediction match the simulationís answer?  Were students close?  Is anybody surprised by this result?  Why or why not?  Is the simulationís answer the best answer to the problem?  Can we be sure that this is the answer?  Why or why not? 

METHOD 2 - THEORETICAL PROBABILITY (exact)

13. Can we calculate the probability of at least one match?  What makes calculating this difficult?  Is it easier to calculate a different probability?  Can you suggest another probability we could calculate?  How would you calculate the probability of no matches?  What is the relationship between these two probabilities?

14. What is the probability that the first two people do not share a birthday?

15. Assuming that the first two do not share a birthday, what is the probability that the third person does not share either of their birthdays?  What is the probability that none of the three people share a birthday?

16. Assuming none of the first three people share a birthday, what is the probability that the fourth person does not share any of the first three birthdays?  What is the probability that none of the four people share a birthday?

17. Can we generate a formula for the kth person not sharing any of the first k-1 birthdays?

18. What is the probability that there is at least one birthday match in a group of k people?

19. Plot points for both graphs [1. Probability of at least one match, and 2. Probability of no matches] using a graphic representation.  (Suggestion: Have the x-axis represent numbers of people and the y-axis represent probability.)  The teacher can lead the class as a whole in calculating points or different students can calculate different points on their own and share them with the class.

20. What do the graphs indicate is the answer to the problem?  Discuss this result.  Did it match the result from the spreadsheet simulation?  Why do you think this happened?

21. Questions to ask about the graphs:

A. (Algebra I - Calculus) Should these graphs be continuous or discrete?  Why?  Why is the first graph increasing?  Why is the second one decreasing?

B. (Algebra II - Calculus) What will the resulting graph look like if we added these two plots together?

C. (Precalculus & Calculus) Calculate the point at which the first graph is increasing the fastest.  At which point is the second graph decreasing the fastest?  How are the derivatives of the two graphs related?

METHOD 3 - OPPORTUNITIES FOR MATCHES

22. If there are two people in the room, how many possible birthday matches are there?  What is the probability that this match would not occur?

23. How many possible birthday matches are there for three people?  Are these possible matches independent events?  What is the probability that none of these matches would occur?

24. Repeat this for four and five people.

25. Can we write this probability as a general equation?

26. Why are the probabilities from this method different from those in Method 2?  What assumption is different?

27. What does this method indicate is the answer to the problem? Discuss the result.  Is it the same as the results from the other two methods?

VII. Assessment:

1. The different predictions you made for the answer should be listed on your answer sheet.  For each prediction, tell me why you thought that would be the answer.  There is no right or wrong answer here.  What was your reaction to the answers we derived in class?

2. Use Methods ____ to answer the following questions: How many people must be in a room before the probability that some share a birthday becomes at least 90 percent?  99 percent?  (The individual teachers should decide which method(s) they want their students to use.)

Extension #1. What is the probability of exactly two people in a room of fifty sharing a birthday?  In other words, what is the probability that there are forty-nine distinct birthdays in a room of fifty people?

Extension #2. Connection to History/Government: Look up the birthdays of all of the Presidents of the United States.  Are there any shared birthdays?  What is the probability that this would occur?

Extension #3. Look up the birthdays of twenty-three celebrities of your choice.  Were there any matches?

The Birthday Problem Solution Guide

References:

Lesser, L.M.  (1999).  Exploring the birthday problem with spreadsheets, The Mathematics Teacher (92), No. 5 pp. 407-411.

Stewart, C.D.  (1999).  Exploring the Birthday Problem.