Odds of Having The Same Birthday
by It is more likely than you might think to find a shared birthday in a small group.
Odds of Having the Same Birthday
Let’s make a bet. I will bet you even money that if you take 30 people selected at random, at least two of them have the same birthday. To avoid making things more complicated than they need to be, we will eliminate Feb 29 as a possible birthday. Anybody with that day as a birthday will be considered as Mar 1. So we will be dealing with 365 possible birthdays. 365 possibilities among 30 people sounds like we are unlikely to have a match. You want to take the bet? Good. Put your money down. I am grinning and will still be grinning when I collect my winnings.
We can approach this problem from either of two directions: the likelihood that two people have the same birthday or the likelihood that all birthdays are different. Let’s look at it. The math is not difficult and I will do it for you. Consider the case of two people. We are told the first person’s birthday. What are the odds the second person’s birthday is different? There are 364 days to choose form, so the odds are 364/365. Better than 99%.
Okay, we now have two people with different birthdays. What happens when we add a third? This one has odds of 363/365 to have a different birthday. Still over 99%.
Well, then with 30 people, the 30th person would have odds of 336/365, right? Over 92%. Sounds like your side of the bet is rock-solid. Not so.
When you consider the likelihood of any one of a number of independent outcomes, the result is computed not by adding the probabilities but by multiplying them. The reason for this is that you are looking for the probability that event A does not happen AND event B does not happen AND event C does not happen, and so forth. Without getting too involved, let me state the probability at least two of 30 randomly selected people share a birthday is about 70%. It is a pretty safe bet—for me.
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