Expected number of distinct birthdays in a room of 30

Question

Thirty people are in a room; assume each person's birthday is uniform over $365$ days and independent of the others.

What is the expected number of distinct calendar days that appear as someone's birthday?

Accepted Answer

Let Id indicate that day d is somebody's birthday, so the number of distinct birthdays is D = \sum{d=1}^{365} Id. A given day d is missed by one particular person with probability \frac{364}{365}, and missed by all 30 independently with probability \left(\frac{364}{365}ight)^{30}. So: E[Id] = 1 - \left(\frac{364}{365}ight)^{30}. By linearity of expectation (the days are dependent, but linearity does not care): E[D] = 365\left(1 - \left(	frac{364}{365}ight)^{30}ight). Estimate the power with \left(	frac{364}{365}ight)^{30} \approx e^{-30/365} = e^{-0.082} \approx 0.921, so E[D] \ap…

Expected number of distinct birthdays in a room of 30

Your answer

More Probability questions