The famous Birthday Problem was presented in Example 2.22. Now suppose you have 500 Facebook friends. Make the same assumptions here as in the Birthday Problem.

a. Write a program to estimate the probability that, on at least one day during the year, Facebook tells you three (or more) of your friends share that birthday. Based on your answer, should you be surprised by this occurrence?
b. Write a program to estimate the probability that, on at least one day during the year, Facebook tells you five (or more) of your friends share that birthday. Based on your answer, should you be surprised by this occurrence?

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Example 2.22 The Birthday Problem. Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days. If ten people are randomly selected, what is the probability that all have different birthdays? Imagine we draw ten days, with replacement, from the calendar to represent the birthdays of the ten randomly selected people. One possible outcome of this selection would be (March 31, December 30, ...., September 27, February 12). There are 365¹0 such outcomes. The number of outcomes among them with no repeated birthdays is (365) (364) (356) = 365 P10 (any of the 365 calendar days may be selected first; if March 31 is chosen, any of the other 364 days is acceptable for the second selection; and so on). Hence, the probability all ten randomly selected people have different birthdays equals 365P10/365¹0 = .883. Equivalently, there's only a .117 chance that at least two people out of these ten will share a birthday. It's worth noting that the first probability can be rewritten as 365P10 36510 365 364 365 365 356 365