How do you calculate the probability of *only* 2 people having different birthdays (excluding leap years)?

Question

I know that the numerical answer is $\frac{364}{365}$, but I didn't understand why. this answer seems to contradict my current understanding of probability. For example, to calculate the probability of rolling the same number on $2$ dice, the formula is $\frac{6}{6 \times 6}$ ; the denominator is the total possible outcomes on one die multiplied by the total possibilities on the other, and the numerator is the total number of favourable outcomes. However, using this logic the probability of two people having the same birthday would be $\frac{132496}{133225}$ (which is $\frac{364 \times 364}{365 \times 365}$ rather than $\frac{364}{365}$. So where am I making my mistake?

Answer 1

The numerator is 365*364. There are 365 choices for the birthday of the first person, and for each of these there are 364 choices for the birthday of the second one.