I came across the birthday problem on a forum and I found that there were two answers. My knowledge of probability is pretty limited so I can't quite understand the difference.
X and Y having the same birthday (ignoring the leap year case):
Case 1.
1-(365/365 * 364/365)
I am yet to make sense of the birthday paradox and there seem to be a lot of variations where there are 1,2,...n people in a room, at a party and so on. I won't ask for the explanation here but I have one question: Why take the complement instead of calculating the answer directly? (I read some explanations but they were too mathematical and all I got was that it was difficult to do so. What's the intuition behind it?)
Case 2.
1/365
In my understanding, for X to have been born on the same day as Y, there is only one outcome, 1 in 365 days. So the answer is 1/365. (Does this case come under the birthday paradox?)
In Case 1, people seem to influence each other's probabilities (as opposed to Case 2 where Y's birthday was sort of constant?) but there seems to be some nuance to the wordings which makes these two problems different. What am I missing here?