From James Nugent comes a puzzle that goes out to all the folks with March birthdays:
Earlier today, James’s boss was surprised to find out that not only did no one on their team have a birthday this week, but that nobody was celebrating a birthday for the entire month. With a total of 40 people on the team, the probability of this happening seemed to be miniscule.
But was that really the case? What was the probability that none of the 40 people had birthdays this month? (For the purpose of this riddle, assume that a year consists of 12 equally long months. It’s a sufficiently good approximation!)
Extra credit: What is the probability that there is at least one month in the year during which none of the 40 people had birthdays (not necessarily this month)?
A randomly selected unknown person has an $\dfrac{11}{12}$ probability of not having a birthday this month.
None of the 40 employees having a birthday this month is
$$\left( \dfrac{11}{12}\right)^{40} \approx 0.0308$$Using bars and stars, the total number of ways 40 people's birthdays can be distributed over 12 months is: $${40+12-1 \choose 12-1}$$
In order to solve the extra credit, use complementary counting on the opposite, that is, assume that each month has at least one birthday. That leaves 28 people to be distributed over 12 months:
$${28+12-1 \choose 12-1}$$The answer is:
$$1 - \dfrac{\dfrac{39!}{28!11!}}{\dfrac{51!}{40!11!}}$$$$ = 1 - \dfrac{39!40!}{28!51!}$$$$ = 1 - \dfrac{39\cdot38\cdot\cdot\cdot 29}{51\cdot50\cdot\cdot\cdot 41} \approx 0.9648$$