With the Six Nations Rugby Championship underway, we posed a puzzle about birthdays and related this to the size of rugby union teams and squads, as follows.
First of all think about how many people there need to be in a group for the probability of at least two people sharing a birthday to be greater than 50%. Or put another way how big does the group have to be before you will wager your hard-earned cash on two or more people sharing a birthday.
Working our way through the team and squad sizes . . .
Do you place a bet on a single rugby union team of 15 people?
What about the match-day squad of the 15 players and eight substitutes – 23 in all?
Maybe you are tempted by the combined starting fifteens of 30 players?
Or perhaps the combined match-day squads of 46 players.
Last offer – the two squads and the four match officials (referee, two assistant referees and the imaginatively named fourth official, a total of 50 individuals.
Still not placed a bet? Then, as we asked before, how many people would you need before opening your wallet?
The way to solve this is to calculate the opposite – what is the chance of the group not sharing a birthday. As follows . . .
The probability of two people not sharing a birthday is 364 / 365, ignoring for a moment Leap Years.
Adding a third person, the probability of them not sharing a birthday with either of the other two is 363 / 365. And combining with the probability of the original two not sharing a birthday, the total for three people is (363 * 364) / (365 * 365).
Adding the fourth, fifth, sixth etc. people follows the same pattern.
When this sum is less than 0.5 the probability of this number of people having unique birthdays is below 50%. And it follows that at this point the probability of any two (or more) sharing a birthday must now be above 50%.
Surprisingly the size of the group when this happens is 23, the size of a rugby union match-day squad. In fact, for the 50 people in the last suggestion the probability of two or more people sharing a birthday has risen to over 97% - almost printing money.
Why do people guess numbers much higher than the actual answer? Our theory is that you subconsciously image yourself sharing a birthday with someone, rather than all of the individuals with each other.