The Coupon Collector's Dilemma in Card Drawing

The classic problem of the Coupon Collector's Problem has long been a fascinating topic in probability theory. However, when we apply this concept to card drawing from a standard deck, it adds an additional layer of complexity due to the interdependence of card draws. In this article, we will explore the expected number of cards drawn to obtain at least one card from each of the four suits in a deck of 52 cards. We will delve into the solution using the coupon collector's framework and the challenges introduced by the dependencies between draws.

Introduction to the Problem

A standard deck of playing cards contains 52 cards, divided equally into four suits: hearts, diamonds, clubs, and spades. The problem is to determine the number of cards that one must draw, on average, to ensure that at least one card from each suit is obtained. This is a variant of the classic coupon collector's problem, which typically assumes that each draw is independent and identically distributed (IID).

Steps to Solve: Understanding the Problem

Let's break down the problem into steps:

Initial State: You start with zero cards from any suit. Goal: Collect all four suits. First Suit: The expected number of draws to get the first suit is always 1, since any card drawn will be a new suit. Subsequent Suits: The probabilities change with each draw, as the composition of the remaining deck affects the likelihood of drawing a card from a particular suit.

Expected Draws for Each Suit

Collecting the First Suit

The expected number of draws to get the first suit is:

E1 1

Collecting the Second Suit

After collecting the first suit, the probability of drawing a new suit is 39 out of 52. The expected number of draws is:

E2 52 / 39 ≈ 1.3333

Collecting the Third Suit

With two suits already collected, the probability of drawing a card from one of the remaining suits is 26 out of 52. The expected number of draws is:

E3 52 / 26 2

Collecting the Fourth Suit

When three suits are already obtained, the probability of drawing a card from the last remaining suit is 13 out of 52. The expected number of draws is:

E4 52 / 13 4

Total Expected Draws

The total expected number of draws to get one card from each suit is calculated as the sum of the expected draws for each step:

E E1 E2 E3 E4 1 1.3333 2 4 ≈ 8.3333

Conclusion: A More Challenging Version of the Problem

This version of the problem is more challenging than the classic coupon collector's problem because of the dependencies between draws. In the classic problem, each draw is independent and identically distributed, but here, once a card is drawn, it changes the probability of drawing a card from another suit in the subsequent draws.

The expected number of draws for the classic coupon collector's problem with 4 suits is 8.3333, which is the harmonic sum of the number of cards to be drawn for each suit. In this case, due to the dependencies, the expected number of draws is slightly smaller than this value. Indeed, it has been derived that the true expected number of draws is approximately 8.3333 - 2/3 or about 7.4444. This small difference is relatively insignificant given the large number of cards in the deck, as expected.

Understanding and solving such variations of the coupon collector's problem not only enhances our grasp of probability theory but also provides insight into the complexities of real-world scenarios where dependencies play a crucial role.