This is Problem 2.2 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition.
You go to a party with 500 guests. What is the probability that exactly one other guest has the sam birthday as you? Calculate this exactly and also approximately by using the Poisson PMF. For simplicity, exclude birthdays on February 29.
My first thought was that p should be $(1/365)^2$. The idea: p is the probability that event A is X date and B is that same X date.
$P(A \cap B)$ for independent events = $P(A) * P(B)$ = $(1/365) * (1/365)$
My second thought was that p should be (1/365). The idea: p is the probability of B is X date given A is also X date.
$P(A | B)$ with conditional probability = $P(A) * P(B|A) = 1 * 1/365 = 1/365$
I assigned 1 to P(A) since my birthday is given and is certain to be on that date.
How do you determine what p is?
