By likely, we mean greater than 50% chance. With one person there is a 0 percent chance that you'll have two people with the same birthday. With two people the probability that they won't share a birthday is 364/365. The probability that they will share a birthday is therefore 1 - (364/365). With three people the probability that they won't share a birthday is the same as for two people, times 363/365. So the probability that three people will share a birthday is 1 - (364/365) * (363/365). Notice that with each additional person added, the probability that they shares a birthday with one of the previous persons goes up, because there are fewer "free" days remaining. We keep adding people until the %age is greater than 50%. When we have 23 people the %age is 50.729%.
