Birthday Coincidence

Compute the complementary probability that all 23 birthdays are distinct.

Multiply the sequential probabilities $\frac{365}{365}\cdot \frac{364}{365}\cdots \frac{343}{365}$.

1. 1. Complement

   It's easier to compute the probability all birthdays are distinct.

2. 2. Sequential probability

   Assign birthdays one at a time; the $k$-th new person avoids all previous distinct birthdays with probability $(365-k+1)/365$.

   $$P(\text{distinct})=\prod _{k=0}^{22}\frac{365-k}{365}$$

3. 3. Final answer

   Subtracting from 1 gives $P\approx 0.5073$.

   $$P=1-\prod _{k=0}^{22}\frac{365-k}{365}\approx 0.5073$$

Answer: 1 - product from k=0 to 22 of (365-k)/365