Problem
A group of 50 people are comparing their birthdays (as usual, assume their birthdays
are independent, are not February 29, etc.). Find the expected number of days in the year on which at least
two of these people were born.
Solution
By linearity of expectation, the answer is 365 times the probability that at least two people were born on a given day. For a given day, there are
50 choose 2 or 1225 ways to choose two people who are born on that day and since the remaining people could be born on any day there are 365^48 choices for their birthdays. Dividing by 365^50, the number of possibilities with no restrictions, and multiplying by 365 yields, the expected number of days on which at least 2 people were born is
365(1225*365^48/365^50) = 1225/365, which is incorrect.
What is wrong about this approach?
Best Answer
I think the linearity of expectation approach is fine, but like I said in my comment, you didn't compute the probability of $2$ or more people sharing a given birthday correctly.
Take January $1$ as the birth date. This is a binomial distribution question. We can get the probability of $0$ people having that birthday by doing:
$\binom{50}{0}\cdot(\frac{1}{365})^{0}\cdot(\frac{364}{365})^{50}\approx0.8718$
And the probability of one person being born on that date as
$\binom{50}{1}\cdot(\frac{1}{365})^{1}\cdot(\frac{364}{365})^{49}\approx0.1198$
Then we get the probability of $2$ or more January $1$ birthdays by subtracting:
$1-(0.8718+0.1198)\approx0.0084$
Finally, we take your suggestion of linearity of expectation to get the answer:
$365\cdot(0.0084)\approx3.0757$ which is different from your value of $\frac{1225}{365}\approx3.356$