Problem 72 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A list of poker hands ranked in order from the highest to the lowest is shown in the following table, along with a description and example of each hand. Use the table to answer. If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: Four of a kind?

Step-by-Step Solution

Answer

There are 624 different hands that consist of Four of a Kind in a well-shuffled 52-card deck.

Step 1: Recognize the cards' distribution in Four of a Kind

In a Four of a kind hand, there are four cards of the same rank, and one card of a different rank. There are 13 distinct ranks (Aces, Kings, Queens, etc.) and 4 different suits (Spades, Hearts, Diamonds, and Clubs) available in a deck of 52 cards.

Step 2: Choose the rank for Four of a Kind

First, we will choose the rank for the four cards with the same value. There are 13 different ranks that can be chosen, so we can use the combination formula to select one rank out of 13: \[\binom{13}{1} = 13\]

Step 3: Choose the suits for Four of a Kind

Now, we need to pick the four cards of the same rank, which means we need to choose their suits. There are 4 suits available, and we will pick all 4, so it is: \[\binom{4}{4} = 1\]

Step 4: Choose the rank for the fifth card

We have already chosen the rank for the four cards with the same value, and now we need to choose the rank for the fifth card. There are 12 remaining ranks (excluding the rank we used for Four of a Kind), so we use the combination formula again: \[\binom{12}{1} = 12\]

Step 5: Choose the suit for the fifth card

Lastly, we need to determine the suit for the fifth card. There are four suits to choose from, so the combination formula gives us: \[\binom{4}{1} = 4\]

Step 6: Calculate the total number of Four of a Kind hands

Now, we just need to multiply the combinations from the previous steps to get the total number of Four of a Kind hands: \[13 \times 1 \times 12 \times 4 = 13 \times 12 \times 4 = 624\] So, there are 624 different hands that consist of Four of a Kind in a well-shuffled 52-card deck.

Key Concepts

Combination Formula

The combination formula is a fundamental tool in probability and combinatorics, which allows us to calculate the number of ways to choose a certain number of items from a larger set, without considering the order. The formula is given as \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \( n \) is the total number of items, \( k \) is the number of items to choose, and \( ! \) represents the factorial operation (e.g., \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)).

In the context of poker hands such as Four of a Kind, this formula is essential. To create a Four of a Kind, one must select 4 cards of the same rank; the combination formula simplifies the process of determining how many unique combinations can be made from the available cards. For example, to choose the rank of the Four of a Kind, you would apply the combination formula as \( \binom{13}{1} \), because from 13 different possible ranks, you're selecting one. Although choosing all four suits seems trivial (as there's only one way to do it), the formula can still express this action as \( \binom{4}{4} = 1 \). It's this methodical application of the combination formula that enables accurate calculations of possible hands.

Probability in Card Games

Probability plays a crucial role in card games, especially in games like poker where the outcome is strongly influenced by the likelihood of getting a specific hand. Probability in card games is usually expressed as a fraction or percentage that indicates how likely it is to draw a hand out of all possible hands.

For example, in the calculation of the Four of a Kind hand in poker, the number of desirable outcomes (Four of a Kind hands) is divided by the total number of possible 5-card hands from a deck of 52 cards. To find the total, we use the combination formula \( \binom{52}{5} \), which provides the full range of potential hands. Knowing the probability of obtaining a Four of a Kind is useful for players strategizing their moves based on the risks and rewards of different poker hands. It also illustrates how mathematical concepts directly influence games of chance and strategy, guiding decisions in games that might seem purely luck-based at a glance.

Applied Mathematics

Applied mathematics is the branch of mathematics that deals with methods and techniques used to solve real-world problems in disciplines such as science, engineering, business, and economics. One engaging application of applied mathematics is in the analysis of games and gambling.

Understanding poker hand probabilities, for example, requires combinatorics to account for the various ways cards can be combined, as well as probability theory to gauge the chances of forming specific hands. The precise calculation of 624 different Four of a Kind hands is only possible through the application of mathematical principles and formulas. This effective utilization of mathematics not only enhances our appreciation for the complexities of seemingly simple card games but also demonstrates the value and practicality of mathematical concepts in everyday activities and decision-making processes.