How many weekdays hold a birthday?

Question

Three people each have a birthday that falls on one of the 7 days of the week, independently and uniformly at random. On average, how many distinct weekdays carry at least one of the three birthdays?

Accepted Answer

Instead of enumerating how the three birthdays might collide, put one indicator on each of the 7 weekdays. Fix a weekday. It carries no birthday only if all 3 people were born on a different day. Each person avoids a given weekday with probability 	frac67, so P(	ext{a given weekday is empty}) = \left(	frac67ight)^3 = \frac{216}{343}. Then a given weekday holds at least one birthday with probability 1 - 	frac{216}{343} = 	frac{127}{343}. Let Xi = 1 if weekday i is used; E[Xi] = 	frac{127}{343}. Summing over 7 weekdays, E[X] = 7 	imes \frac{127}{343} = \frac{127}{49} \approx 2.592. Tak…

How many weekdays hold a birthday?

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