In Blitzstein's Introduction to Probability, it is stated that the probability that any two people have the same birthday is 1/365. However, isn't this the conditional probability that the second person has the same birthday as the first person, given the birthday of the first person?
Couldn't I prove this by simply generalizing the formula that no n people have the same birthday:
$\ P = 1 – (364 / 365)^n$
And simply call n = 2, so that I can interpret the probability of having at least 2 people with the same birthday as the probability that 2 people have the same birthday? This would not be 1/365.
Is my reasoning flawed?
Best Answer
Unfortunately, yes, there is flaw. According to your purported formula, the probabilty of having two people with the same birthday, when you only have $n=1$ person, is:
$$P_1 = 1 - \Big( \frac{364}{365} \Big)^1 = 1 - \frac{364}{365} = \frac{1}{365} \neq 0.$$
So, you are ascribing a non-zero probability to an impossible event. Have a think about whether that is the correct formula, and what kind of change you might make to it.