Coupon collector's problem

In probability theory, the coupon collector's problem describes the "collect all coupons and win" contests. It asks the following question: Suppose that there is an urn of n different coupons, from which coupons are being collected, equally likely, with replacement. What is the probability that more than t sample trials are needed 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 ${\displaystyle \Theta (n\log(n))}$. For example, when n = 50 it takes about 225 trials to collect all 50 coupons.

Let T be the time to collect all n coupons, and let t_i be the time to collect the i-th coupon after i − 1 coupons have been collected. Think of T and t_i as random variables. Observe that the probability of collecting a new coupon is p_i = (n − (i − 1))/n. Therefore, t_i has geometric distribution with expectation 1/p_i. By the linearity of expectations we have:

Here H_n is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain:

where ${\displaystyle \gamma \approx 0.5772156649}$ is the Euler–Mascheroni constant.

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