Question

Two decks of cards are "matched," that is, the order of the cards in the decks is compared by turning the cards over one by one from the two decks simultaneously; a "match" means that the two cards are identical. Show that the probability of at least one match is nearly \(1-1 / e.\)

Short answer

The probability of at least one match is nearly \(1 - \frac{1}{e}\).

Step-by-step solution

Understand the Problem

Two decks of cards are turned over one by one simultaneously, and a match occurs when the two cards are identical. We need to determine the probability of at least one match.

Identify the Total Number of Cards

Each deck contains 52 cards. Therefore, there are 52 comparisons to be made.

Use the Principle of Independent Trials

Each card showing is an independent event, and the probability of not matching any of the cards in 52 trials can be approximated using the principle of derangements in probability theory.

Define Derangements

A derangement is a permutation of elements such that no element appears in its original position. For 52 cards, calculate the probability of no matches (derangements).

Calculate the Probability of No Matches

The probability of no matches in a derangement for a large number of items (n cards) is given by \(\frac{1}{e}\). Here, since there are 52 cards, the derangement probability is \(\frac{1}{e}\).

Calculate the Probability of At Least One Match

The probability of at least one match is the complement of the no matches probability. Therefore, it is given by \(1 - \frac{1}{e}\).

Conclusion

Hence, the probability of at least one match when comparing two shuffled decks of 52 cards is nearly \(1 - \frac{1}{e}\).

Key concepts

Derangements

To understand the core concept of derangements, first consider what happens when you try to rearrange items such that no item appears in its original position. Think of it as shuffling a set of elements so that no element ends up in its starting spot. This is crucial in the context of matching cards from two decks.

When comparing two decks, each card has a twin that is its match in the other deck. A derangement ensures that no card in one deck matches its position in the other deck. For 52 cards, the number of derangements can get quite complex, but the underlying principle is about generating permutations where none of the elements align with their original positions.

The probability that a derangement occurs (i.e., no card matches) for a large number of items, like 52 cards, approaches \(\frac{1}{e}\), where \('e'\) is Euler’s number, approximately 2.718.

Permutations

Permutations refer to the different ways in which a set of items can be arranged. For example, if you have a deck of 52 cards, there are \(!52\) (52 factorial) possible ways to arrange the cards. This huge number represents all the possible sequences you can create by shuffling your cards.

When we discuss permutations in the context of derangements, we’re looking at a specific subset of permutations. Ensure that no element (or card) is in its original position. Understanding permutations helps in grasping why derangements are a special case and how their probability is derived.

In the case of matching cards, understanding permutations allows us to count the total number of possible arrangements (52 factorial), and then use this to identify how many of these arrangements are derangements (where no card aligns).

Independent Events Analysis

Independent events occur when the outcome of one event does not affect the outcome of another. In the context of our card matching problem, turning over a card from one deck and comparing it with the corresponding card in another deck are treated as independent trials.

Each pair of cards in the two decks can either match or not, independently of what the previous pairs of cards did. This makes analyzing the problem simpler, as the probability calculations for each pair can be performed separately without worrying about interdependence.

By treating each turn of a card as independent, we can apply the principle of derangements to find the overall probability of no matches, and subsequently, the probability of at least one match through simple calculations.

Combinatorial Probability

Combinatorial probability refers to the likelihood of an event occurring based on the possible combinations of outcomes. In the case of card matching, we look at all the possible ways to arrange and compare the decks.

The key here is to evaluate how many of these arrangements lead to matches and how many do not. By comprehensively analyzing these combinations, one can derive probabilities accurately. Here, we focus on the specific combinations that result in derangements.

To find the probability of no matches (derangements), we calculate how many of the possible permutations of the deck result in no matches and compare it to the total number of permutations. Since calculating exact numbers for large sets like 52 cards can get cumbersome, we rely on the approximation that for large 'n', the probability of no matches is \(\frac{1}{e}\).

Consequently, the probability of at least one match is deduced to be very close to \(\text{1 - }\frac{1}{e}\).