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The Birthday Paradox

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Abstract

This project shows how mathematical probability sometimes contradicts our intuition. Despite the fact that there are 365 days in a year, if you survey a random group of just 23 people there is a 50:50 chance that two of them will have the same birthday. Don't believe it? Try this project and see for yourself.

Summary

Areas of Science
Difficulty
 
Time Required
Very Short (≤ 1 day)
Prerequisites
None
Material Availability
Readily Available
Cost
Very Low (under $20)
Credits

Objective

Investigate whether or not the birthday paradox holds true by looking at random groups of 23 or more people.

Introduction

The birthday paradox, also known as the birthday problem, states that in a random gathering of 23 people, there is a 50% chance that two people will have the same birthday. Is this really true?

Due to probability, sometimes an event is more likely to occur than we believe it to, especially when our own viewpoint affects how we analyze a situation. For example, one reason why the birthday paradox seems like a paradox is that when in a room with 22 other people, if a person compares his or her birthday with the birthdays of the other people it would make for only 22 comparisons, or, in other words, only 22 chances for people to share the same birthday. But when all 23 birthdays are compared against each other, it makes for much more than 22 comparisons. How much more? While the first person has 22 comparisons to make, the second person was already compared to the first person, so there are only 21 comparisons to make. The third person then has 20 comparisons, the fourth person has 19 comparisons, and so on. If you add up all possible comparisons (22 + 21 + 20 + 19 ... + 1) the sum is 253 comparisons, or combinations. Consequently, each group of 23 people involves 253 comparisons, or 253 chances for matching birthdays. But how do 253 comparisons lead to a 50% chance of two people having the same birthday? Check out the resources in the Bibliography below to help you find out how!

In this science fair project, you will investigate whether the birthday paradox holds true by looking at several random groups of 23 or more people. Will 50% of the groups include at least two people with the same birthday, making the birthday paradox hold true?

Terms and Concepts

Questions

Bibliography

There are a number of different sites that explain the Birthday Paradox and explain the statistics. Here are a few to get you started:

Materials and Equipment

Experimental Procedure

  1. First you will need to collect birthdays for random groups of 23 or more people. Ideally you would like to get 10-12 groups of 23 or more people so you have enough different groups to compare. (You do not need the year for the birthdays, just the month and day.) Here are a couple of ways that you can find a number of randomly grouped people:
    1. Most schools have around 25 students in a class, so ask a teacher from each grade at your school to pass a list around each of his/her classes to collect the birthdays for students in each of his/her classes.
    2. Use the birthdays of players on major league baseball teams. (Note: This information can easily be found on the internet). Alternatively, use the birthdays of other random people using online sources.
  2. For each group of 23 or more birthdays that you collected, sort through all the them to see if there are any birthday matches in each group. How many of your groups have two or more people with the same birthday? Based on the birthday paradox, how many groups would you expect to find that have two people with the same birthday? Did the birthday paradox hold true?
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Variations

  • In this science project you used a group of 23 or more people, but you could try it using bigger groups. If you use a group of 366 people (the greatest number of days a year can have) the odds that two people have the same birthday are 100% (excluding February 29 leap year birthdays), but what do you think the odds are in a group of 60 or 75 people?

Careers

If you like this project, you might enjoy exploring these related careers:

Career Profile
Statisticians use the power of math and probability theory to answer questions that affect the lives of millions of people. They tell educators which teaching method works best, tell policy-makers what levels of pesticides are acceptable in fresh fruit, tell doctors which treatment works best, and tell builders which type of paint is the most durable. They are employed in virtually every type of industry imaginable, from engineering, manufacturing, and medicine to animal science, food… Read more
Career Profile
Life is full of risks to both people and property. Actuaries predict the chances that future negative events will occur in a person's life, and then think of ways to reduce those chances, or reduce the impact of those negative events. Actuaries help bring peace of mind to both families and to businesses. Read more

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Cite This Page

General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.

MLA Style

Science Buddies Staff. "The Birthday Paradox." Science Buddies, 20 Nov. 2020, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p007/pure-mathematics/birthday-paradox?from=Home. Accessed 3 Dec. 2022.

APA Style

Science Buddies Staff. (2020, November 20). The Birthday Paradox. Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p007/pure-mathematics/birthday-paradox?from=Home


Last edit date: 2020-11-20
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