The Wikipedia article about the birthday problem explain how to calculate the probability of birthday collision. They're simply telling in the process that the events are independent :
When events are independent of each other, the probability of all of the events occurring is equal to a product of the probabilities of each of the events occurring.
I don't understand how they're independent. If one person enter the room, the probability that his birthday is different than the previous person is 1 (100%). If the second person enter the room, the probability that his birthday is different than the previous person is 364/365 (99.73%). Therefore, aren't the events dependent?
I know that the event is independent if $P(A \cap B) = P(A) P(B)$. I know the probability of both events (1 and 364/365), but how do I calculate the probability of both elements happening? Saying that I just need to use that formula because they're independent wouldn't make sense since that's what I'm trying to figure.
Best Answer
The important issue with the birthday problem is that each person's BIRTHDAY is independent. Your point that the chance of collisions increases as the number of people in the room increases is exactly the point of the calculation.
An example of a case where birthdays are not independent would be, testing the chance of collision at a twin convention.
The way I think about the solution is to take the inverse of the case where everybody has different birthdays. When two people are in the room, the odds of no collision are $\frac{365}{365} * \frac{364}{365}$, so the odds of a collision are $1-\frac{365}{365} * \frac{364}{365}=\frac{1}{365}$. When a third person enters the room, the odds of no collision are $\frac{365}{365} * \frac{364}{365}* \frac{363}{365}$, so the odds of a collision are $1-\frac{365}{365}*\frac{364}{365}*\frac{363}{365}=\frac{1093}{133225}=1-\frac{\frac{365!}{(365-n)!}}{365^n}$