# The Birthday Paradox

Difficulty | |

Time Required | Very Short (≤ 1 day) |

Prerequisites | None |

Material Availability | Readily Available |

Cost | Very Low (under $20) |

## 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.## Objective

Investigate whether or not the birthday paradox holds true by looking at random groups of 23 or more people.## Share your story with Science Buddies!

Yes, I Did This Project! Please log in (or create a free account) to let us know how things went.## 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*, 28 July 2017, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p007/pure-mathematics/birthday-paradox. Accessed 13 Dec. 2018.

### APA Style

*The Birthday Paradox.*Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p007/pure-mathematics/birthday-paradox

Last edit date: 2017-07-28

## 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

- Birthday paradox
- Probability

### Questions

- What is the probability that a coin flipped will land "heads"?
- What is the probability that a coin flipped three times in a row will land "heads" each time?
- What are the odds that two people share the same birthday in a group of 366 people? (
*Hint*: 366 is the greatest number of days a year can have.) - If 253 unique comparisons can be made between 23 people, how does this lead to a 50% chance of two people in a group of 23 having the same birthday?
(
*Hint*: Check out the resources in the Bibliography below.)

## 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:- Azad, K. (n.d.).
*Understanding the Birthday Paradox*. Retrieved March 3, 2012, from http://betterexplained.com/articles/understanding-the-birthday-paradox/ - Math is Fun: Advanced. (n.d.).
*Combinations and Permutations.*Retrieved March 3, 2012, from http://www.mathsisfun.com/combinatorics/combinations-permutations.html - Wikipedia contributors. (2012, March 13).
*Birthday problem.*Wikipedia, The Free Encyclopedia. Retrieved March 3, 2012, from http://en.wikipedia.org/w/index.php?title=Birthday_problem&oldid=481710309

## News Feed on This Topic

*Note:*A computerized matching algorithm suggests the above articles. It's not as smart as you are, and it may occasionally give humorous, ridiculous, or even annoying results! Learn more about the News Feed

## Materials and Equipment

- Birthdays for random groups of 23 or more people (ideally 10–12 groups). See the Experimental Procedure
- Lab notebook

The Birthday Paradox

#### www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p007/pure-mathematics/birthday-paradox

## Experimental Procedure

- 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:
- 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.
- 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.

- 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?

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

### Statistician

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Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering. Read more### Actuary

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- 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?

## Share your story with Science Buddies!

Yes, I Did This Project! Please log in (or create a free account) to let us know how things went.## Ask an Expert

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