Maths for Mums & Dads

The Risk of Floods

Return periods and birthday coincidences


The recent floods in Shropshire were described as a 'One in 100 year' event.  Civil engineers call this the Return period - meaning you would only expect an event like this to return once in 100 years.

Unfortunately as often happens with situations involving probability, the return period is often misinterpreted.  The most common mistaken beliefs are:

*  This event will happen exactly once every 100 years.

*  If the event hasn't happened for 100 years then one must be "due" now.

I was recently invited onto a podcast called Engineering Matters to explain the maths of return periods.  I used the analogy of birthday coincidences.

Imagine bumping into some random person in the street.  What is the chance that you and they share the same birthday?  Ignoring leap years, the chance is 1 in 365.

Now imagine 365 people are lined up in front of you.  You ask each person in turn to tell you their birthday.  How long will it be before you meet somebody with your birthday?  It's possible that you might have to wait until the 365th person, but it's actually very likely it'll happen earlier.  The odds are about 63% (nearly two thirds) that somebody before person 365 has your birthday (the calculations are at the bottom of this blog).  In fact there's a 10% chance that the coincidence will happen before you get to the 40th person.

But conversely, there is a decent chance that not only will you fail to find somebody in the first 365 people, you might not even meet somebody with your birthday in the first 1000 people that you encounter. The chance that none of the first 1000 people you meet has your birthday is 6%, a bit more than 1 in 20.  Gosh!

What does this mean for Return periods?  A flood might be a 1 in 100 year event, but it probably won't be 100 years before the event happens again.  If your local river flooded recently and they said it was a 1 in 100 year occurrence, then you might want to invest in some sandbags.  There's a 10% chance it could happen again in the next decade. And that's before you allow for any impact that global warming might have. 

Probability can be very counter-intuitive.

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LISTEN HERE to the Engineering Matters episode about flooding, my section begins about 25 minutes in:  

https://engineeringmatters.reby.media/2020/03/05/49-uncharted-waters-flooding-and-drought/ 

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THE CALCULATIONS

The easiest way to work out the probability that you'll meet somebody with your birthday in the first N people you encounter, is to work out the chance that you WON'T meet somebody with your birthday, and subtract that from 1.  The chance that none of N people will share your birthday is (364/365)^N.

     N                 Chance no-one has your birthday (P)        Chance at least one person does (1-P)

     10                                 97%                                                              3%

      40                                90%                                                              10%

      100                              76%                                                              24%

      365                              37%     (~1/e)                                                63%

      1000                              6%                                                              94%