It’s that day of the year again, when florists make half their yearly income from selling flora to those in, or freshly out of, love. I proud myself on having at least had email or some other form of internet conversation with my small contingent of FB friends. There’s only 150 odd of them in total. And yet, 2 of them have a birthday on Valentine’s Day, Feb 14 ! What are the odds ?
Well, the odds are actually pretty good, and I used that as a party trick on a quiet night shift once, we counted everyone, from ambulance officer, to patient, to ward clerk to policeman coming through the door to find 2 people with the same birthday, and it took only 36 or so to find them. How is this possible ? Well, it’s called the Birthday paradox, and it is based on the inability of our brains to think in exponentials, and to consider scenarios that we are not personally involved in:
It’s like asking “What’s the chance of getting one or more heads in 23 coin flips?” There are so many possibilities: heads on the first throw, or the 3rd, or the last, or the 1st and 3rd, the 2nd and 21st, and so on.
How do we solve the coin problem? Flip it around (Get it? Get it?). Rather than counting every way to get heads, find the chance of getting all tails, our “problem scenario”.
If there’s a 1% chance of getting all tails (more like .5^23 but work with me here), there’s a 99% chance of having at least one head. I don’t know if it’s 1 head, or 2, or 15 or 23: we got heads, and that’s what matters. If we subtract the chance of a problem scenario from 1 we are left with the probability of a good scenario.
The same principle applies for birthdays. Instead of finding all the ways we match, find the chance that everyone is different, the “problem scenario”. We then take the opposite probability and get the chance of a match. It may be 1 match, or 2, or 20, but somebody matched, which is what we need to find.
Fact of the matter is that you need just 23 people to have a 50:50 chance to find two with the same birthday. Which is why I now better go and send some virtual flowers to my two FB friends whose birthday it is today !