## Thursday 7 March 2019

### At least two pupils in the same school class will have the same birthday - probably! (the 'Birthday Paradox')

Our old Maths teacher always used to do this trick with every new class intake each year...

First, he would ask the question 'What are the chances that at least two pupils in this class have the same Birthday'?

After asking for our guesses, he would ask anyone in the class to raise their hand if they had a Birthday in January. Then he would ask each of the pupils who had raised a hand to shout out the day they were born on, e.g.Tom shouts '21st', Victor shouts '12th', etc. and then he would go on to February, etc. until someone else shouted out 'YES' if they had the same Birthday.

Theoretically, with a class size of 30 pupils, there is a 70% chance that two or more pupils in the same class will share the same Birthday (and probably the same birth-year as well). Our Maths teacher would tell us that he would explain this when we came to study probabilities, but he also said something else which peaked our curiosity...

He said that he had been doing this test for many years now and with many classes, and that he has found that the probability was actually much better than the theoretical 70% figure!

He would then ask if anyone could think of why this should be?

The basic probability calculation is not trivial but is explained on wikipedia  here.

It can be seen that with a sample class size of 30 we get an approximate 70% probability of two or more people having the same birth day.

So why should my teachers 'real life' results give a far greater probability than 70%...?

### Reason 1: Births are 'seasonal'

Given any specific year -  e.g. a class born in 2004-2005 for instance, there may have been a very cold spell (causing more 'early nights') in that year, cold Winter nights, power cuts on certain days, and, of course, holidays would occur within the same specific dates on the Calendar, etc. These 'events' usually cause a small peak in births 9 months later, for reasons which should be obvious!

The 'heat chart' below shows that as soon as the colder weather starts (Oct-Jan including Xmas) then there are more births 9 months later in June-September. It also shows that, contrary to folklore, Spring is not the most common time for 'making' babies or giving birth (perhaps due to our modern living habits?).
 Click on Source to view the interactive version which shows the probable conception date for each square.

### Reason 2: Weekends and Bank Holidays

Many births are 'natural' and as such will tend to occur 'randomly' throughout the  week. However, some births are induced (usually by giving an injection). In other cases, a Caesarean Section or other surgical procedure may need to be performed.

Due to hospitals having fewer staff and facilities available at weekends, holiday periods and Bank Holidays, there are always less births during these periods (a doctor would not induce a birth on a Sunday unless it was necessary, he\she would wait until Monday). Christmas and July 4th (in the USA) always have fewer births (and more births just before and after these days)! There are more births in September (which is 40 weeks after the XMas holidays).

These reasons make it even more likely that a school class of 30 pupils (who were all born in the same year) will include at least two people who share exactly the same birth date!