First, if this question is too basic for math.stackexchange, I apologize. I wasn't sure where else to ask, but if you have a suggestion I'll happily take the question elsewhere.
I'm totally mathematically unsophisticated, and I was asked a theoretical question I don't know how to answer. The question is:
Given: I want to invite as many people as possible to my birthday party. I do not want people who have the same birthday as me to attend. Everyone I invite who does not share my birthday will attend. People who do share my birthday will be jealous and only have a 1 in 3 chance of attending the party if they are invited. How many invitations should I send if I want there to be no more than a 50% chance of someone with the same birthday as me showing up?
Now, I tried to approach this as follows:
The chance that a random other person would have the same birthday as you is 1/365. The chance that someone who has the same birthday as you would accept the invitation is 1/3. Therefore the combined probability of someone having the same birthday and choosing to attend is 1/1095.
Every time another person attends we are adding one more chance of the above happening, so we can think of this as:
0.5 = n/1095
n = 0.5 / (1/1095) = 547.5
Well, I was told that the above is not correct, but the reason why I was wrong and the correct approach to understanding this problem was not explained to me.
Could anyone explain my mistake and how to correctly calculate this probability problem? Thanks!