How many calendar days stay birthday-free?

Question

A company of 500 employees has birthdays independent and uniform over 365 days. On average, how many calendar days are nobody's birthday?

Accepted Answer

Handle one day at a time and add up, using linearity of expectation. A given day stays empty only if all 500 people miss it: P(	ext{day empty}) = \left(\frac{364}{365}ight)^{500} \approx 0.253. Give each of the 365 days an indicator that equals 1 when it is empty. By linearity, the expected number of empty days is 365 times that chance: E[	ext{empty days}] = 365 	imes \left(\frac{364}{365}ight)^{500} \approx 365 	imes 0.253 \approx 92.6. So on average about 92 to 93 days go unclaimed, even with 500 people. The clustering of birthdays (collisions) is exactly why 500 people fail to cove…

How many calendar days stay birthday-free?

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