Problem 85 - A First Course in Probability

Question

There are \(k\) types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type \(i\) with probability \(p_{i}, \quad \sum_{i=1}^{k} p_{i}=1\) If \(n\) coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of \(n\) coupons.)

Step-by-Step Solution

Answer

The expected number of distinct types of coupons that appear in a set of n collected coupons can be found using the formula \(E(X) = \sum_{i=1}^{k} (1 - (1 - p_i)^n)\), where k is the number of coupon types, and \(p_i\) is the probability of collecting a type i coupon.

Step 1: Define the random variable X

Let's define the random variable X as the number of distinct types of coupons out of the n collected coupons. We want to find the expected value of X, E(X).

Step 2: Calculate the probability of not obtaining a specific type of coupon

To find the probability of not obtaining a specific type of coupon (let's say coupon i), we need to consider the complement event, i.e., the probability that we never collect coupon i in the n collected coupons. As each new coupon collected is of type i with probability \(p_i\), then the probability of not collecting a type i coupon in a single draw is \(1 - p_i\). Since the draws are independent, the probability of not obtaining type i coupon in n collected coupons is \((1-p_i)^n\).

Step 3: Calculate the probability of obtaining a specific type of coupon

Since we have calculated the probability of not obtaining a specific type of coupon, now we need to find the probability of obtaining at least one of type i among n collected coupons. This probability is the complement of the probability calculated in step 2. So, the probability of obtaining a specific type of coupon, say coupon i, is given by \(P_i = 1 - (1 - p_i)^n\).

Step 4: Calculate the expected value of X

Our goal is to find the expected number of distinct types that appear in a set of n coupons collected. We can use the linearity property of expected value and the fact that the expected value of each coupon type being in the collection is the same as the probability calculated in step 3. The expected value can be calculated as follows: \[E(X) = \sum_{i=1}^{k} P_i = \sum_{i=1}^{k} (1 - (1 - p_i)^n)\] We just need to plug in the values for the probabilities and the number of coupons collected to find the expected value of X.

Step 5: Solve for the specific case

To solve for a specific case, we just need the values of k, n, and the probabilities for each type of coupon. With these values in hand, we can use the formula derived in step 4 to find the expected number of distinct types that appear after n coupons are collected.

Key Concepts

Expected Value

When you hear 'expected value', think of it as the average outcome you would expect after many trials of a random process. In probability theory, the expected value is a key concept that represents the average of all possible outcomes, each weighted by its probability of occurrence. It provides a measure of the center of the distribution of a random variable. For example, if you were rolling a six-sided die, the expected value of the roll would be 3.5, because over a large number of rolls, the average of the results would approach this value.

In the coupon collector's problem, each type of coupon has a different probability of being collected, denoted as \(p_i\) for the i-th type. To calculate the expected number of distinct types appearing after collecting \(n\) coupons, we use the formula:
\[E(X) = \.\sum_{i=1}^{k} P_i\]
where \(E(X)\) is the expected value of distinct coupon types collected and \(P_i = 1 - (1 - p_i)^n\) is the probability of collecting at least one of the i-th type of coupon.

Application in Solving the Problem

For the coupon collector's problem, you first need to calculate the chance of collecting each type of coupon at least once (\(P_i\)). Then, since the scenario involves multiple types of coupons and each type's presence in the collection is an independent event, you sum these probabilities to find the expected number of distinct types of coupons, \(E(X)\).

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and the likelihood of events occurring. It's based on the idea that even in processes where the outcome is uncertain, there are patterns and structures that can be understood and predicted mathematically. In our daily lives, we often interpret probability informally as the chance of something happening or not.

In the context of the coupon collector's problem, probability theory provides the framework for modeling the collection of coupons as a series of independent trials, each with their own chance of success (collecting a distinct type of coupon). Probability theory allows us to calculate not only the likelihood of collecting each type of coupon but also to establish the relationship between these probabilities and the overall expected outcome (the total distinct types collected).

Underlying probability theory is a set of axioms that ensure consistency in our calculations; for instance, probabilities cannot be negative and must sum up to one across all possible outcomes for a given event. In the example provided, we confirmed that the sum of all probabilities for each type (\(\sum_{i=1}^{k} p_{i}\)) equals one, satisfying one of these axioms.

Independent Events

In probability theory, events are considered independent if the occurrence of one event has no effect on the likelihood of the occurrence of another event. This means the odds of one event happening remain constant regardless of whether another event has occurred.

Understanding independent events is crucial for solving the coupon collector's problem. Here, the collection of each coupon type is an independent event. When you collect a coupon, whether you've collected other types of coupons has no influence on the type of the new coupon. This means the probability of finding a particular type of coupon every time you collect one remains unchanged — a fundamental characteristic of independent events.

Application in Solving the Problem

The assumption of independence simplifies the calculations in the coupon collector's problem significantly. Since the events (collecting a specific type of coupon) are independent, we can simply multiply the individual probabilities of not collecting each type to find the probability of not having it in the n collected (\((1-p_i)^n\)). The complement of this — the probability of having collected the coupon at least once — is key in finding our expected value.