Q: How many people do I need to have at my birthday party so that there is a better-than-even chance that one of them will share My birthday??…
Recall that famous Birthday Problem asks how many people you need to have at a party so that there is a better-than-even chance that any two of them will share the same birthday. To figure out the exact probability of finding two people with the same birthday in a given group, it turns out – as many times in math – to be easier to ask the opposite question.
What is the probability that NO two will share a birthday, i.e., that they will all have different birthdays?
Let´s see, step by step!: With just two people, the probability that they have different birthdays is 364/365, or about .997. If a third person joins them, the probability that this new person has a different birthday from those two (i.e., the probability that all three will have different birthdays) is (364/365) x (363/365), about .992. With a fourth person, the probability that all four have different birthdays is (364/365) x (363/365) x (362/365), which comes out at around .983. And so on. The answers to these multiplications get steadily smaller. When a twenty-third person enters the room, the final fraction that you multiply by is 343/365, and the answer you get drops below .5 for the first time, being approximately .493. This is the probability that all 23 people have a different birthday. So, the probability that at least two people share a birthday is 1 – .493 = .507, just greater than 1/2.
Now our problem here is a bit different…
A: Disregarding leap years, there are 365 possible birthdays, so there’s a 1 in 365 chance that any random person will have the same birthday as mine.
However, there’s not quite a 2 in 365 chance that at least one of two randomly invited persons will have the same birthday as mine. That chance is actually 1 – (364/365)^2 (that is, the complement of the chance that both of the two people have a different birthday from mine) which is just slightly less than 2/365.
If I invite 4 guests, the chance will be about 0,01091 = 1.091% that one or more will have the same birthday as mine. That’s the closest I can get to a 1% chance.
To get closest to a 10% chance I have to invite more than 36.5 people (and how do I invite half a person anyhow?) Inviting 37 people gives me a 9.653% chance (1 – (364/365)^37) but 38 people gets me closest, with a 9.900% chance.
To get closest to a 50% chance I have to invite 253 people. (50.048% chance)!!…