About 50 people need be present for the 'coincidence' of two of them having the same birthday to become, roughly, a 30-1 on chance. In a company of 100 employees the odds are more than three million to one on that two share a birthday. When the first person enters the room and announces their birthday, the probability of the second person sharing the same birthday is 1/365. Conversely, the probability of the second birthday being different is the opposite of the first calculation, 364/365. When two birthdays are known, the probability of the third being different is 363/365, as there are now two 'favourable' outcomes among 365. The compound probability of birthday 2 being different from birthday 1, and of birthday 3 being different from the other two, these being independent outcomes, is:- (364/365)*(363/365) = 0.991796 or 99.2% chance that two people will not share the same birthday. Note the start of the sequence is (365/365). We have removed this as it does not affect the result of the calculation. All that is necessary now is to continue adding terms to the fraction until it equals less than 1/2 or 50%, since as soon as the probability is less than 1/2 that all birthdays are different, the probability is clearly more than 1/2 that any two are the same. In other words it is more likely than not that two people in the room share the same birthday.

First, let us say that we have 'b' amount of people in a room, and that none of them are twins or born on leap years. Also, let's assume that every day of the year is just as likely a time as any other one to be born. And let's say we want to find how many of these people have the same birthday. We go about this by first calculating what the probability is of them not having the same birthday. So let's imagine these people all line up, and they come up to us and we ask when their birthday is. When we find out, we write it down on a blackboard. Let's find the probability of a match, which we'll call p(b), probability of birthday. 'b' is the number of people in the room The first person has a 100% chance of having a birthday that's not up on the board, cuz hell, there are non up there yet. The next person comes up and has at 1 (decimal version of 100%) - 1/365 chance of having a birthday not yet seen on the board. The third person in line has a 1 - 2/365 chance of having a birthday that's not seen yet. And so on. And so on. To calculate the probability, multiply 1(1-1/365)(1-2/365)...[1-(b-1/365)] *[1-(b-1/365)] shows that not all sequences in the term are there, and that it goes on until we get to one less than the amount of people, 'b', we have in the room.* So then, after we find that number, we can take the compliment of the number we just got, p(b). A compliment is easy to find, all it is is 1 minus the number. 1 - p(b). It expresses the opposite of the statement we already had...so it poses the question: "What's the probability someone in the room has the same birthday?" And as it turns out, that probability is quite high. With 10 people, the a 12% chance someone will have the same birthday. With 20, 41%. With 30, 70%. And with 50, 97%

Nobody is answering the question asked. "Is it true that if you're in a room with 23 people you have around a 50% chance of sharing a birthday with one of them?" This is not asking what the probability of any two people sharing a birthday but what is the chance of YOU sharing a birthday. To get a 50% chance to THAT question you need 183 people. 365/2 = 182.5 so 185 will give you a slightly better chance.