I am working on the problem of birthday paradox:
How many people do you need to reach 50% chance of at least one pair with matching birthday?
When I first heard the problem I verified the answer 23 by subtracting possibility of 23 people with different birthdays from 1.
$$ 1 - \frac{{}_{365} \mathrm{P}_{23}}{365^{23}} $$
I assumed this should work as people are distinguishable objects and 23 people have ${}_{365} \mathrm{P}_{23}$ permutations of not matching birthdays.
And yet according to scientific american article one should calculate
$$ 1 - \left(\frac{364}{365}\right)^{\sum_{ i = 1 }^{22} i}$$
because we make $\sum_{ i = 1 }^{22} i$ comparisons and two people have $\frac{364}{365}$ chance of not matching birthday.
Am I missing something here?
The first method is exact. The article's method is an approximation. What you are looking for is the Reverse Birthday Problem, which has a nice approximation.