Suppose that there are n guests at a Halloween party, and that each is wearing one of 200 possible costumes available at local store, uniformly at random and independently of all other guests. Using Poisson approximation and the value log2 0.83, show that only about n = 17 guests are needed to ensure that some pair of guests are wearing the same costume with probability at least 50%.

Note: This approximation is quite accurate. It can be shown that, when n = 17, the true probability of a match is 1 (200)₁₇/200¹⁷ 50.3%