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 Birthday Paradox

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!


  1. I was in Grade 4 in 1965 when Mrs. Benham told us that in our class of 22 (back then, that was the class size) it was 50-50 that two of us shared the same birthday. She explained that in any group of more than 21, it was as likely, as not that a birthday would be shared. I started laughing. I sat right in front of Mrs. Benham and she got a little cross that I was interrupting the lesson. I kept giggling as she decided to 'prove it' by asking us our birthdays.

    Still giggling, I huffed out my birthday and waited for the girl right behind me, my neighbour Susan, to admit to having the SAME BIRTHDAY!! But even before she could say that, the THIRD student in the row, a girl named Nancy, blurted out, "Hey, that's MY birthday!!"

    So, Mrs. Benham got her proof, by asking ONE student and finding THREE students with the same birthday. Oh, and two other kids in the class shared the same birthday too. Probability be darned.

    When I got to high school, the biggest social event of the calendar was Dec. 28th, a birthday shared by the event host and FOUR other members of the class, plus one kid from the next youngest class who played up a year with the basketball team. Not QUITE a group of 22 or 30, but still pretty good proof of the non-randomness of birthday spread across the 365.25 days of the year.

    Thanks for the software,thanks for the interesting post. GM

  2. Yep, 28th Dec is a 'hot' day - as the staff don't want to induce or operate over the XMas holidays if they can help it. It can also depend on what day of the week XMas happens to be in any one year as thus affects what is the first fully operational day back at work. I knew a couple who were doctors and they planned their two sons to be born on the same day but one year apart and induced the 2nd so they could have just one Birthday party on the same day!