Here's the answer to the riddle I posted a few days ago.
Imagine a soccer field with 23 people on it, two teams of 11 players and the referee. What is the probability that any two of those 23 people share the same birthday?
With 23 people and 365 birthdays to choose from, it would seem highly unlikely that anybody would share the same birthday.
In fact the answer is just over 50%, 50.7% to be exact – that is to say, on the balance of probability, it is more likely than not that two people on the field will share the same birthday.
The reason for this high probability is that what matters more than the number of people is the number of ways people can be paired. When we look for a shared birthday, we need to look at pairs of people, not individuals.
Whereas there are only 23 people on the field, there are 253 pairs of people (23 x 11). For example, the first person can be paired with any of the other 22 people, giving 22 pairings to start with. Then the second person can be paired with any of the remaining 21 people (we have already counted the second person paired with the first person so the number of possible pairings is reduced by one), giving an additional 21 pairings. Then the third person can be paired with any of the remaining 20 people, giving an additional 20 pairings, and so on until we reach a total of 253 pairs.
There are more steps to calculating the exact probability, but already with 253 pairs and 365 possible birthdays, it does not seem unreasonable that the probability of a shared birthday is significant.
This seems intuitively wrong, and yet it is mathematically undeniable. Strange probabilities such as this are exactly what bookmakers and gamblers rely on in order to exploit the unwary.