In probability theory, the coupon collector's problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? An alternative statement is: Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once? The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons.
