(a) Consider the following variation of the coupon...

(a) Consider the following variation of the coupon collector's problem. Each box of cereal contains one of $2 n$ different coupons. The coupons are organized into $n$ pairs, so that coupons 1 and 2 are a pair, coupons 3 and 4 are a pair, and so on. Once you obtain one coupon from every pair, you can obtain a prize. Assuming that the coupon in each box is chosen independently and uniformly at random from the $2 n$ possibilities, what is the expected number of boxes you must buy before you can claim the prize? (b) Generalize the result of the problem in part (a) for the case where there are $k n$ different coupons, organized into $n$ disjoint sets of $k$ coupons, so that you need one coupon from every set.

(a) Consider the following variation of the coupon collector's problem. Each box of cereal contains one of $2 n$ different coupons. The coupons are organized into $n$ pairs, so that coupons 1 and 2 are a pair, coupons 3 and 4 are a pair, and so on. Once you obtain one coupon from every pair, you can obtain a prize. Assuming that the coupon in each box is chosen independently and uniformly at random from the $2 n$ possibilities, what is the expected number of boxes you must buy before you can claim the prize?
(b) Generalize the result of the problem in part (a) for the case where there are $k n$ different coupons, organized into $n$ disjoint sets of $k$ coupons, so that you need one coupon from every set.

Key Concepts

Uniform Random Sampling

The assumption that every coupon is equally likely to be drawn in each trial, simplifying analysis by ensuring that the random process is memoryless and that each draw is identically distributed. This concept is critical for calculating probabilities and expected values in the context of collecting items.

Expected Value Calculation

A fundamental concept in probability that involves calculating the average number of trials required to achieve a given outcome by summing over all possible scenarios weighted by their probabilities. This plays a central role in determining how many boxes of cereal one must buy to claim the prize in both the classic and modified coupon collection scenarios.

Coupon Collector's Problem

This classical problem in probability theory addresses how many independent random trials are expected in order to collect a complete set of distinct items. It introduces key ideas such as expectation, probability distributions, and harmonic series, and is a foundational model for studying random sampling and collection processes.

Group-Based Collection

An extension of the coupon collector’s problem where the items are partitioned into groups, and the goal is to collect at least one item from each group rather than every unique item. This variation modifies the classical approach by grouping outcomes and redefines the collection criteria, which in turn alters the expected number of trials needed.

Transcript

00:01 In this exercise, we simulate the coupon collector problem.

00:06 The scenario here is, cereal boxes come with one coupon each, and there are 10 unique coupons that may exist in each cereal box.

00:18 For each cereal box, each of the coupons is equally likely to be in it, but each box only comes with one coupon.

00:27 And so we are interested in the number of cereal boxes that we must purchase in order to collect all 10 coupons.

00:36 Now i've produced some code in r, which looks like this.

00:42 Now i've produced some code in r, which looks like this.

00:52 And the way the code works is i represent the 10 unique coupons with a vector of the numbers 1 through 10.

01:01 And then i start off each replication by sampling 10 from the number of coupons.

01:07 So that represents buying the first 10 boxes.

01:11 Because we know we need at least 10 boxes in order to collect all 10 coupons.

01:14 Coupons.

01:17 And then i check to see if there exists any coupon that is not in the boxes that we already have.

01:24 So that is this statement here.

01:29 It's the statement is true if any of the coupons does not exist in our collection of boxes.

01:37 So while this is true, we keep sampling one more cereal box or one more coupon and append that to our list of cereal boxes that we have or the list of coupons that we've accumulated from the cereal boxes.

01:58 And we simulate this 10 ,000 times.

02:01 So once this condition is not satisfied, that's when we break the wild boop.

02:07 And that means that we have all 10 coupons.

02:11 So we run the simulation 10 ,000 times.

02:14 And for each replication, we append the number of cereal boxes that we had to buy in order to collect all 10 coupons.

02:28 For part a, we are asked the probability of collecting all 10 coupons in the first 10 cereal boxes.

02:39 So if x is the random variable, that is the number of cereal boxes that we had to collect in order to collect all 10 coupons, we're looking for the probability that x is equal to 10, which will be approximately equal to, or hopefully approximately equal to our estimate for this probability based on our simulation...

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