Probability of having the same birthday without complement

Question

I understand the probability of getting at least n amount of people.Assume 365 normal days

But I would someone to check if my reasoning for this particular question is right ?
I would like to know the probabiltiy that 12 people have the same birthdayP(W). I am aware that we can find the complement P(W), but is this also a correct thinking ?

We know the probability of one person having a birthday on any day is 1/365.
Another person A having the same birthday would then be (1/365)^2 because A needs to match that of the first person to be considered the same birthday
And so, like this 12 people would then give you (1/365)^12 ?

But, this is very much wrong and I am not sure where my logic fails ?

Answer 1

We know the probability of one person having a birthday on any day is 1/365.

This means that for any particular day, say Jan 1, the probability that a person having a birthday on that day is $1/365$. Similarly, the probability that $12$ people are born on a particular day like Jan 1 is $(1/365)^{12}$. However, you want the probability that $12$ people have the same birthday, where that birthday could be any of the $365$ days of the year. There are $365$ days, and for each day there is a probability of $(1/365)^{12}$ that the $12$ people will all have that day as their birthday. We add up the probability probabilities of these disjoint events to get the probability of at least one occurring. The result is $365\cdot (1/365)^{12}=(1/365)^{11}$.