Since there are 365 days in the year, the probability of any two people not having the same birthday is $\frac{364}{365}$, the probability of a third person having a different birthday than the other two people is $\frac{363}{365}$, and so on. Thus, in a room containing 23 people, the probability of no two people sharing the same birthday will be:
$\frac{364}{365}\xb7\frac{363}{365}\xb7..\.\xb7\frac{365-22}{365}equals;\frac{364\xb7363\xb7..period;\xb7\left(365-22\right)}{{365}^{22}}$
$\=\frac{365\xb7364\xb7363\xb7..\.\xb7343}{{365}^{23}}\=\frac{\frac{365\!}{342\!}}{{365}^{23}}\approx 0.5$ or $50$%.
If this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is simply $1-0.5\=0.5$ or 50%.