compute the probability that at least 2 have the same birthday?

Question

In a party of 5 persons compute the probability that at least 2 have the same birthday(month/day),assume a 365-day year.

Answer 1

1. Calculate the probability that each birthday is unique:

   • Choose $5$ unique birthdays: $\displaystyle\binom{365}{5}$
   • Multiply by the number of permutations for $5$ people: $\displaystyle\binom{365}{5}5!$
   • Divide by the total number of birthday-combinations for $5$ people: $\displaystyle\frac{\binom{365}{5}5!}{365^5}$

2. Calculate the probability of the opposite event, where at least $2$ birthdays are not unique:

   • $\displaystyle1-\displaystyle\frac{\binom{365}{5}5!}{365^5}=2.7$%

---

Finally, as a little exercise, find the minimum number of people for which the probability that at least $2$ have the same birthday is more than $50$% (and realize that it's less than what you'd expect).