If you are collecting a set of different coupons or trading cards by random chance, how many do you need to collect on average before you have a complete set? This problem has connections to random sampling and is applicable in various collecting scenarios.
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The problem you have described is known as the Coupon Collector's Problem. It can be applied to various collecting scenarios, including collecting coupons, trading cards, or any other type of items randomly distributed.
To find the expected number of items you need to collect before obtaining a complete set, we can start by considering the scenario with two different types of coupons or cards. Let's assume that there are two distinct types: Type A and Type B.
In order to complete the set, you need to collect at least one of each type, which we can refer to as a "coupon pair."
The probability of obtaining a Type A coupon on any given draw is 1, since you have not collected it yet. Similarly, the probability of obtaining a Type B coupon on any given draw is 1. Therefore, the probability of obtaining a coupon pair on any given draw is:
P(coupon pair) = P(Type A) * P(Type B) = 1 * 1 = 1
This means that you are guaranteed to get a coupon pair on each draw until you have collected both types. However, you may also collect duplicates of the same type in the process.
Now, let's extend the problem to include more types of coupons or cards. Let's say there are n distinct types in total.
The probability of obtaining a specific type of coupon on any given draw is 1/n, since you have not collected it yet. Therefore, the probability of obtaining a coupon pair on any given draw is:
P(coupon pair) = P(Type A) * P(Type B) * P(Type C) * ... * P(Type n) = 1 * 1/2 * 1/3 * ... * 1/n
To find the expected number of draws to obtain a complete set, we need to sum the probabilities of obtaining a coupon pair on each draw until we have collected all n types. Mathematically, this can be represented as follows:
1 + 1/2 + 1/3 + ... + 1/n
This expression represents the harmonic series, which does not have a simple closed-form solution. However, it can be approximated using the natural logarithm:
E(number of draws) ≈ n * ln(n)
Therefore, on average, you would need approximately n * ln(n) draws to obtain a complete set of n different types of coupons or trading cards.
It's important to note that this is an expected value and may vary in practice. Some sets may be completed in fewer draws, while others may require more.
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