What are the chances of sharing a birthday?

Chris Smith asked mathematician, Boby Seagull, to breakdown the birthday paradox for listener Liz. 

Bobby - You might think that the most recognised song in the world might be a Stormzy rap or a Taylor Swift song, or Ed Sheeran who seems to be everywhere all the time. But actually, according to the Guiness Book of Records, it’s Happy Birthday which is the most recognised song in the English language. So it’s only appropriate that the birthday paradox is something which is important to maths.

So the question is: what’s the minimum number of people required, let’s say in a room, for the chance of two people sharing the same birthday be more than 50/50, and obviously it means the same day and month, not the year.

Chris - Shall we ask the crew? Giles, what do you think?

Giles - 50/50?

Bobby - The chance of two people in a room having the same birthday being more than 50/50; how many people do you need in a room to guarantee there is more than a 50/50 chance.

Giles - Oh my goodness. About 180 people.

Chris - Chris?

Chris P - About 250.

Rachel - I reckon it’s quite low. Maybe 30.

Bobby - Rachel’s pretty close. Intuitively we seem to think it’s quite a high number but actually it’s 23 people. With 23 people, mathematically, the chances of two people in that room sharing the same birthday is slightly more than 50%.

Chris - Are you going to show your working as all good students should?

Bobby - We’ll try to. This is one of the things where, with a whiteboard and paper this is quite easy to demonstrate, but without we’ll try. A good analogy is…

Chris - I’m going to write this down as you go.

Bobby - Let me just give you an analogy first before you start. One way to think of it is imagine a 365 sided dice and after 23 throws you’re more than likely than not to get two of the same numbers the dice land.

Chris - Where does the 23 number come from?

Bobby -  The 23; we’re about to get there now. Step one: the probability of two people sharing the same birthday in a group is 1 minus the probability of no-one sharing the same birthday. So we’ve got that yet?

Chris - Right.

Bobby - So it’s 1 minus the probability of no-one sharing it. So let’s work out the probability of no-one sharing the same birthday. In a group of 2 people, firstly it’s 365 out of 365, that’s essentially the first person can be born on any day. Then you multiply that by 364 out of 365. I’ll explain the second fraction. That second person can be born any day apart from the first day that the person…

Chris - That you’re born in, yeah?

Bobby - Yeah. So that’s the 364 choices; that’s for 2 people. If we expand it to 3 people now we’ve got one less option so that original multiplication we multiply that 363 over 365. If you keep on doing that, adding 362 over 365, all the way to 23 people at this stage you get this multiplication to be 0.493. If you cast your mind back a minute…

Chris - You wanted 50/50?

Bobby - Yeah. So it’s 1 minus that. So once you get to 23 people, the chance of....

Chris - 51%?

Bobby - Yeah.

Chris - Rachel’s going to dispute your maths now.

Rachel - No, I’m not. I’m just going to point out I’m not nearly that clever but I am basically an engineer by training. So my pragmatic version of answering the question is that I know that in a typical school class, you quite often get two kids sharing the same birthday, so therefore 30 was a good guess from that point of view.

Chris - That’s the benefit of wisdom see.

Rachel - So no maths really, just common sense.

Bobby - On the birthday paradox, if anyone’s a football fan here?

Chris - Giles, you look like a footballer. Are you not a football fan?

Giles - American football fan.

Chris - Chris, are you a footballer?

Chris P - No.

Bobby - Can you indulge me on my football related birthday paradox?

Chris - Go on then.

Bobby - We’ve got the World Cup coming up and this is a great test ground for the birthday paradox. Because, coincidently, the number of people in every World Cup squad is 23 people and there are 32 squads in the World Cup. So if people want to test if this theory is true, I think the World Cup squads get announced on the 4th June. Go on the FIFA website that day and check up all the squads - be a nerd like me - and see how many squads. And in the last two World Cups, I think in the 2014 World Cup there were 16 squads out of 32 that had 2 people sharing the same birthday. And the World Cup before, there were 15 out of 32, so just under 50%. So it does work.

Chris - I’m guessing, but I think people will be looking at the FIFA website for reasons other than who’s got a birthday in common.