# Do Rare Events Happen?

I'd be very surprised if I had three children all with the same birthday, but I read in the paper that it happened to one family...

*This resource is part of the collection Probability and Evidence.*

**Do you think there is a family in the UK with three children who all have the same birthday (but born in different years)?**

How rare do you think this is?

How often do you think a baby is born in the UK who shares their birthday with both their older siblings? Do you think it happens:

- To several families per year?
- To one family per year?
- Only every few years?
- Less often?

**Can you calculate the probability that this might happen to a particular family?**

When you've thought about this question, click on the button below:

A child is born. Later on, some siblings are born...

What is the probability that the second child is born on the same day as the first child?

What is the probability that the third child is born on the same day as the first child?

Can you combine these probabilities to find the probability that all three children are born on the same day?

**What assumptions have you made in these calculations? Do you think these assumptions are reasonable?**

**If there are 1 million families in the UK with at least three children, how many of these would you expect to have three children born on the same day?**

How does this compare with your original thoughts?

*You can read more about the probabilities on Plus or Understanding Uncertainty, and you might like to try Last One Standing.*

Dylan from Colchester Royal Grammar School said

I think this should happen to several families per year, because of the chances of people sharing a birthday.

So, the first baby to be born can have their birthday on any day of the year.

The second baby has a chance of $\frac{1}{365}$ of being born on the same day as the first.

Then, the chance of the third baby being born on the same day is $\frac{1}{365}$.

So, you have to multiply the fractions together to work out the chance of them all being born on the same day:

1 x $\frac{1}{365}$ x$\frac{1}{365}$= $\frac{1}{133225}$.

So, if there are a million families in the UK with 3 children then the chances are that roughly 7 or 8 of them have a shared birthday.

Zach also sent in his solution.

He thought about what assumptions he was making. He also considered whether it would make a difference if you specified a date for all siblings to be born on. You can see his full solution here.