Birthday Paradox Exact Threshold and Collision Probability

Scenario

In a room of $n$ randomly selected people, assume each person's birthday is independently and uniformly distributed across $d = 365$ days of the year (ignoring leap years).

Your Task

1. State the exact probability $P(\text{distinct}, n)$ that all $n$ people have distinct birthdays.
2. Using the exponential approximation $1 - x \approx e^{-x}$, derive the analytical threshold $n \approx \sqrt{2 d \ln 2}$ for the collision probability to exceed $50\%$.
3. Compute the exact collision probability for $n = 22$ and $n = 23$, confirming that $n = 23$ is the minimum integer where $P(\text{collision}) > 0.50$.

Constraints

• $d = 365$ days, uniform independent distribution.

Examples For $n = 23$, $P(\text{collision}) \approx 50.73\%$. For $n = 70$, $P(\text{collision}) > 99.9\%$.