What it shows
In a group of just 23 people there is a better than even chance (about 50.7 percent) that two share a birthday, and with 30 people the chance is about 70.6 percent. Collecting birthdays from many class sections shows how often a shared birthday actually turns up.
The science
It is easier to find the chance that nobody shares a birthday. The second person must avoid 1 date out of 365, the third must avoid 2, and so on, so the chance of no match in a group of n is the product 364/365 times 363/365 and so on for n take away 1 factors. For 23 people this product is about 0.493, so the chance of at least one match is about 0.507. The answer feels wrong because people think of matching their own birthday, but there are 253 pairs among 23 people and any pair can match.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Permission from the school to collect day and month of birth only, no year or names
- A survey sheet for each section
- A calculator or a spreadsheet
- Chart paper for the results
Safety
This project uses safe, everyday materials. Still, read the rules below with a parent before you start.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Make a table of predicted probabilities for groups of 10, 20, 23, 30, 40 and 50 by multiplying the no-match fractions on a calculator: about 12, 41, 51, 71, 89 and 97 percent.
Ask each section teacher to let you collect only the day and month of birth of each student, anonymously.
For every section note the number of students and whether at least one shared birthday exists.
Record the group size and the yes or no result for at least 10 sections.
Group the sections by size and work out the observed fraction with a match in each size group.
Plot the predicted curve and the observed points on the same chart.
Discuss any difference: birthdays are not perfectly evenly spread over the year, which makes matches a little more likely than the formula says.
Present the 253 pairs idea on a card: 23 times 22 divided by 2 pairs, which is why matches are common.
What you should see
Most sections of 30 or more students have a shared birthday, and the observed match rates lie close to the predicted curve, with 23 the size at which a match becomes more likely than not.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1How many people are needed for a 50 percent chance of a shared birthday?
23.
2How many people guarantee a shared birthday?
366, by the pigeonhole principle, or 367 counting 29 February.
3Why does the number 253 matter?
It is the number of pairs among 23 people, and each pair has about a 1 in 365 chance of matching.
4What is the chance of no match among 3 people?
364/365 times 363/365, about 0.992, so a match has a chance of about 0.8 percent.
Take it further
Work out how many people are needed for a 50 percent chance that someone shares your own birthday: it is 253, far more than 23.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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