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

  • $\begingroup$ want to find the probability that at least 2 have the same birthday? $\endgroup$ – Liju May 11 '14 at 18:56
  • $\begingroup$ Find the probability that none of them have same birthday and subtract it from 1 $\endgroup$ – avz2611 May 11 '14 at 19:06
  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).


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