### Connecting and challenging the world's brightest young minds

Does anyone in your class share the same birthday? You might think it’s quite rare in a group that size, after all there are 365 dates to choose from! Would it surprise you to know that there’s about a 50/50 chance that there will be ‘birthday twins’ in a group of just 23 people?

Here’s how!

For starters, we know it must depend on the number of people in the room! More people means more birthdays, and more chances for a match. Of course, if there’s just one person in the room, there’s no chance of finding birthday twins!

## Let’s start with the case where there are just two people in a room.

With two people in a room the question simply becomes “What is the chance that any two people share the same birthday?”

We can represent the options using a tree diagram…

Since there are 365 days in a year (generally), and only one way two people can share a birthday, the probability they share their birthday is 1/365. The probability they do not share their birthday is therefore 1 - 1/365 = 364/365.

## What if we add a third person to the room?

Adding a third person adds new possibilities for finding birthday twins in the room. As before, the first two people might share a birthday, but now the third person might share a birthday with either of the first two people!

Here’s a tree diagram showing all the possibilities…
The first branch represents the probability of the first two people sharing (or not sharing) a birthday. The second branch represents the probability of the third person sharing (or not sharing) a birthday with the first or second person.

The probability that the third person shares a birthday with the first is, again, 1/365. Similarly, the probability that they share a birthday with the second person is 1/365. Therefore, the probability that they share a birthday with either of the first two is 1/365 + 1/365 = 2/365.

You might be tempted to think therefore that the probability of finding birthday twins in a room of three people (remembering that the probability of the first two people being birthday twins is 1/365) is 1/365 + 2/365 = 3/365. However, that’s not how tree diagrams (or the conditional probabilities they represent) work.

The probability of the birthday twins being the third person and the first or second is ‘conditional on’ the first two not sharing a birthday. To get the correct probability we need to follow the route along the ‘tree’ and multiply the probabilities.

• Probability of the first two not being birthday twins: 364/365
• Probability of the third being birthday twins with the first or second: 2/365
• Probability of finding birthday twins between the third and first or second person is therefore 364/365 x 2/365 = 0.005

Therefore, the probability of finding a birthday twin in a room of three people is 1/365 + 0.005 = 0.008 or expressed as a percentage, 0.8% (simply multiply by 100 to get the percentage). This represents the first two being birthday twins OR the third being birthday twins with the first or second.

Alternatively, we can simply look at the one route that leads to no birthday twins!

The probability of finding a birthday twin in a room of three people can therefore be found by subtracting the 'no birthday twins' tree-route multiplication from 1 (in probability 1 means complete certainty – if you add up the probability of all options it must sum to 1).

(as expected, in agreement with the other method)

## We can generalise this to any number of people in a room

This is the tree diagram for 5 people in a room…

Therefore, the probability of finding birthday twins in a room of 5 people is
This is the tree diagram for 10 people in a room…

Therefore, the probability of finding birthday twins in a room of 10 people is

…and so on.

## Try it yourself!

Enter the number of people in a room, and find out what the probability of finding birthday twins is!

Number of People:

23

99%