Probability that two people will have the same birthday

Without counting leap years, the answer is exactly $\frac{365×364×363×362×\ldots×334}{365^{30}}\approx0.2936$

Bonus Question: What is the chance that $366$ people do not share the same birthday?

The answer is: $\color{#FFFFFF}\cancel{\frac{146097×365×364×363×362×\ldots×3×2×1×400^{365}×97}{146097^{367}}}$

If 30 people are seated in a room, if we pick one person and say that the probability that the second person in the room will not have the same birthday is 365/365 * 364/365. Similarly the p that the third person will not have the same birthday as the other two is 1 * 364/365 * 363/365. Extending the same reasoning to 30 persons we can see that the P of no two persons having the same birthday is equal to 1 * 364* 363 * 336 / 365^30 = 365 P 30/365^30 = 0.2936

Birthday Problem