The Birthday Paradox
Areas of Science 
Pure Mathematics 
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!
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Last edit date: 20170728
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/understandingthebirthdayparadox/
 Math is Fun: Advanced. (n.d.). Combinations and Permutations. Retrieved March 3, 2012, from http://www.mathsisfun.com/combinatorics/combinationspermutations.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
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Materials and Equipment
 Birthdays for random groups of 23 or more people (ideally 10–12 groups). See the Experimental Procedure
 Lab notebook
https://www.sciencebuddies.org/sciencefairprojects/projectideas/Math_p007/puremathematics/birthdayparadox
PDF date: 20200121Experimental Procedure
 First you will need to collect birthdays for random groups of 23 or more people. Ideally you would like to get 1012 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 moreActuary
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 moreMath Teacher
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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?
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