Suppose three people are in a room. What is the probability that there is at least one shared birthday among these three people? There are a lot of ways there could be at least one shared birthday. Fortunately there is an easier way. We ask ourselves "What is the alternative to having at least one shared birthday?" In this case, the alternative is that there are no shared birthdays. In other words, the alternative to "at least one" is having none. In other words, since this is a complementary event, P(at least one) = 1 - P(none) We will start, then, by computing the probability that there is no shared birthday. Let's imagine that you are one of these three people. Your birthday can be anything without conflict, so there are 365 choices out of 365 for your birthday. What is the probability that the second person does not share your birthday? There are 365 days in the year (let's ignore leap years) and removing your birthday from contention, there are 364 choices that will guarantee that you do not share a birthday with this person, so the Chapter 9 - Probability and Counting 17 probability that the second person does not share your birthday is 364/365. Now we move to the third person. What is the probability that this third person does not have the same birthday as either you or the second person? There are 363 days that will not duplicate your birthday or the second person's, so the probability that the third person does not share a birthday with the first two is 363/365. We want the second person not to share a birthday with you and the third person not to share a birthday with the first two people, so we use the multiplication rule: 365 364 363 (no shared birthday) 0.9918 365 365 365 P and then subtract from 1 to get P(shared birthday) = 1 - P(no shared birthday) = 1 - 0.9918 = 0.0082.