## Riddler collector

**O**nce in a while a fairly standard problem makes it to the Riddler puzzle of the week. Today, it is the coupon collector problem, explained by W. Huber on X validated. (W. Huber happens to be the top contributor to this forum, with over 2000 answers, and the highest reputation closing on 200,000!) With nothing (apparently) unusual: coupons [e.g., collecting cards] come in packs of k=10 with no duplicate, and there are n=100 different coupons. What is the expected number one has to collect before getting all of the n coupons? W. Huber provides an R code to solve the recurrence on the expectation, obtained by conditioning on the number m of different coupons already collected, e(m,n,k) and hence on the remaining number of collect, with an Hypergeometric distribution for the number of new coupons in the next pack. Returning 25.23 packs on average. As is well-known, the average number of packs to complete one’s collection with the final missing card is expensively large, with more than 5 packs necessary on average. The probability distribution of the required number of packs has actually been computed by Laplace in 1774 (and then again by Euler in 1785).

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