➡ The problem:
Hal and Carol accept an invitation to a house party. Hal sees that there will be 30 people attending the party and says to Carol, “I bet you $100 that at least two people will have the same birthday there!”
Should Carol take that bet? (Assume that birthdays are distributed equally over a 365-day year.)
➡ The solution:
This is the famous “Birthday Problem” that never ceases to amaze people. The easiest way to solve it is to consider its complement: For a party of a particular size, what is the probability that no one will share a birthday? If it’s not true that no one shares a birthday, then it must be the case that at least two people share a birthday!
Let’s consider a group of five people. The first person will have some random birthday between January 1 and December 31. The second person, then, will have a probability of (364/365) of having a birthday that is not the same as the first person’s birthday. The third person, then, has 363 “free” dates to choose from, so has a 363/365 probability of having a birthday that is not the same as either the first or the second person’s birthdays. The fourth person will have a 362/365 chance of not sharing anyone’s birthday, and so on.
To then get the probability that not one of the five people will share a birthday, we need to multiply these probabilities together:
P = (364/365)*(363/365)*(362/365)*(361/365) = 0.97
In other words, in a room of five people, no one will share a birthday 97 percent of the time. This seems pretty intuitive. This means there is only a 3 percent chance (100 percent - 97 percent) that at least two people will share a birthday in that group.
What may be less intuitive (at least for me!) is how quickly this number grows. The table below uses the same calculation to show the probability that at least two people will share the same birthday in parties of increasing sizes:
In a party of 30 people, there is a 71 percent chance that two people will share a birthday. Carol should not take that bet!
This problem always brings to my mind some family lore: Years ago, the legend has it that my father mentioned this birthday problem to a woman he was wooing as he was bringing her to a party of about 30 people. She did not believe that the probability could be so high. My dad suggested testing it out and asked her when her birthday is. “May 20!” she exclaimed. It turns out that’s the same birthday as my dad! They never went on another date—which worked out well for me!