Problem 74 - 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: Two pair?

Step-by-Step Solution

Answer

There are 123,552 different 5-card poker hands consisting of two pairs in a well-shuffled deck of 52 cards.

Step 1: Choose two ranks for the pairs

There are 13 different ranks in a deck of cards (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K). To form two pairs, we need to choose 2 different ranks. This can be done in \(C(13, 2)\) ways.

Step 2: Choose the suits for each pair

Now we need to choose the suits for each pair. There are 4 suits (Hearts, Diamonds, Clubs, Spades) and each pair can be formed by selecting 2 suits out of the 4. This can be done in \(C(4, 2)\) ways for each pair. Since we have two pairs, the number of ways to choose the suits for both pairs is \(C(4, 2) * C(4, 2)\).

Step 3: Choose the fifth card

The fifth card must have a rank different from the two pairs. We have 11 ranks left to choose from. For each of these remaining ranks, there are 4 suits to choose from. Thus, we have \(11 * 4\) ways to choose the fifth card.

Step 4: Calculate the total number of two pair hands

To get the total number of two pair hands, we need to multiply the number of ways to choose the ranks, the suits for each pair, and the fifth card. This means: \[ C(13, 2) * C(4, 2) * C(4, 2) * (11 * 4) \] Computing this, we get: \[ \binom{13}{2} \cdot \binom{4}{2} \cdot \binom{4}{2} \cdot 11 \cdot 4 = 123,552 \] Therefore, there are 123,552 different 5-card poker hands consisting of two pairs.

Key Concepts

Poker Probability

Understanding probability in poker can be quite exciting and can significantly improve your gameplay. The concept of poker probability involves calculating the likelihood of certain card hands being dealt. When you know the total set of possible outcomes, which in a standard deck of cards is the 52 cards, you can calculate the probability for various outcomes like getting a two-pair hand.
For example, in calculating the two-pair hand probability, you first determine how many ways these hands can be arranged. This requires considering the number of ways to pick ranks and then suits for these chosen ranks. Finally, identifying how the leftover card—the fifth card—can be selected from the remaining possibilities completes your probability estimation.
By combining these calculated arrangements, you efficiently determine the precise odds of being dealt a specific poker hand, in this case, a two-pair, helping you make more informed decisions during actual poker games.

Card Combinations

Card combinations are essential in understanding how to strategically play poker. In combinatorics, a combination refers to how items (such as cards) are arranged or selected from a larger set. In poker, creating different hands involves calculating combinations.
For a two-pair hand in poker, you begin by choosing two ranks out of the 13 available (A, 2, 3,..., K). This is expressed mathematically as \(C(13, 2)\). Then, for each pair of ranks selected, you need to pick two suits from the four available. This is calculated using \(C(4, 2)\), done twice—once for each pair.
Understanding card combinations not only helps in counting possible poker hands but also provides deep insight into crafting strategies and knowing when to bet or fold.

Mathematical Counting

Mathematical counting in combinatorics is a fundamental skill used to solve problems involving large and complex sets of possibilities, like those found in games of chance. Counting involves identifying the total number of outcomes or arrangements possible from a given set, like a deck of cards.
To calculate the total number of two-pair poker hands, you multiply different calculated probabilities. First, find the number of ways to choose two ranks, then multiply it by the ways to select suits for these ranks. Lastly, factor in the number of ways to choose the fifth, remaining card. This process showcases the systematic way of counting that ensures every possible combination is considered.
Mastering counting techniques can greatly enhance your ability to solve a wide array of mathematical problems beyond just poker, sharpening your logical reasoning and decision-making skills.