The Birthday Problem

You go with a friend to a football game. The game involves 22 players of
the two teams and one referee. Your friend wagers that, among these 23
people on the field, at least two people will have birthdays on the same day.
If this is the case you will have to pay 1 euro to your friend, otherwise he
will pay you 1 euro.

Would you accept the bet? Or equivalently who is more likely to win?

Can be computed by:

1. Combinatorics
   • Probability of $k$ people having different birthdays
   • First person can have a birthday on any day
   • Probability the second person has the same birthday as the first person: $\frac{1}{365}$
   • Probability the third person has the same birthday as the first person or the second person: $\frac{2}{365}$
   • $\therefore P(k)=\frac{365}{365}(1-\frac{1}{365})(1-\frac{2}{365})...$
2. Simulation
   • Assumptions:
     • Ignore leap years
     • Assume all days are equally likely
   • Assign a random birthday to all 23 players
   • Iterate over players and check for duplicates
   • Count this experiment as a fail if a duplicate is found
   • Repeat this experiment a large amount of times and compute the Relative Frequency of fails
   • $p=\frac{fails}{experiments}$
   • This method is known as Monte Carlo simulation