The birthday paradox

Imagine the scene at the Brazilian football team's hotel. Hulk and Paulinho are relaxing after another stylish win. Talk turns from tactics to post World Cup plans, BBC News reports. "It'll be one party after another," says Hulk, confidently assuming Brazilian victory on home soil. "First the World Cup, then my birthday a couple of weeks later." "Your birthday's in July?" replies Paulinho. "Me too - 25 July, when's yours? "No way, exactly the same day!" exclaims Hulk incredulously. "What are the chances of that?" With 365 days in a regular year, most people's intuitive answer would probably be: "Pretty small." But in this case our intuition is wrong - and the proof of that is known as the birthday paradox. "The birthday paradox is one of maths' greatest hits," says Alex Bellos, author of Alex Through the Looking Glass: How Life Reflects Numbers and Numbers Reflect Life. "It's something you can say in one line which gives you this 'wow'!" In its most famous formulation, the birthday paradox says that you only need a group of 23 people for there to be a greater than 50% chance that two of them share the same birthday. (For lovers of detail, we should be clear that by birthday we mean day and month, not year.) Bellos points out that the birthday paradox isn't a logical paradox - there's nothing self-contradictory about it, it's just unexpected. Imagine a typical classroom. "People think it's an amazing coincidence that two people in a class of 30 share the same birthday," he says. "Actually, with 30 people it's a 70% chance." Or consider your favourite social media site. If you've got 70 friends, you've got a 99.9% chance that at least two of them will share a birthday. But perhaps the best data-set of all to test this on is the football World Cup. There are 32 teams, and each team has a squad of 23 players. If the birthday paradox is true, 50% of the squads should have shared birthdays. Using the birthdays from Fifa's official squad lists as of Tuesday 10 June, it turns out there are indeed 16 teams with at least one shared birthday - 50% of the total. Five of those teams, in fact, have two pairs of birthdays. The list is: Spain, Colombia, Switzerland (x2), USA, Iran (x2), France (x2), Argentina (x2), South Korea (x2), Cameroon, Australia, Bosnia Herzegovina, Russia, Netherlands, Brazil, Honduras and Nigeria. One of Argentina's pairs, Fernando Gago and Augusto Fernandez, share the same actual birth date - 10 April 1986. Details also at