Logic
# Paradoxes in Probability

Three friends find out their birthdays are all within the same week.

Supposing their birthdays are otherwise random, what is the probability they all have their birthday on the same day?

Thirty people all have their birthdays in November (which has 30 days).

If their birthdays are otherwise random, what is the probability that none of them share a birthday?

Suppose a computer system locked via a 32-bit password (so there are $2^{32}$ possible numbers). The passwords are generated randomly but stored in such a way that the system is vulnerable if a password is a duplicate of another one.

What's the probability of a password duplicate with 50,000 users?

Minah is trying to solve the problem below:

2 people, Angel and Bob, have birthdays in September (which has 30 days). Assuming their birthdays are otherwise random, what is the probability they have the same birthday?

Minah's argument goes like this: Angel's birthday can be any of the days. We want to know the chances that Bob's birthday matches Angel's. Each day has a $\frac{1}{30}$ chance of being Bob's birthday. So Bob's chance of sharing a birthday with Angel is $\frac{1}{30} .$ Is this argument correct?