Problem 71 - 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: A flush (but not a straight flush)?

Step-by-Step Solution

Answer

There are 5,108 different 5-card poker hands that consist of a flush, but not a straight flush, when dealt from a well-shuffled deck of 52 cards.

Step 1: Calculate total flush hands

To form a flush hand which contains 5 cards of the same suit, we can choose the suit first and then choose the 5 cards from that chosen suit. There are 4 suits in total, and we can choose any 5 cards from the 13 cards in each chosen suit. Therefore, the total flush combinations can be calculated as: \((4 \text{ suits}) \times \binom{13}{5} \text{ (combinations of choosing 5 cards from the 13 cards in each chosen suit)}\)

Step 2: Calculate total straight flush hands

A straight flush is a hand that contains 5 cards of the same suit in sequence. To find the number of straight flush hands, we need to calculate the number of sequential combinations for each suit. There are 4 suits, and 10 possible sequences for a straight flush within each suit (A-2-3-4-5, 2-3-4-5-6, 3-4-5-6-7, ..., 10-J-Q-K-A). The total combinations of straight flush hands can be calculated as: \((4 \text{ suits}) \times (10 \text{ possible sequences for each suit})\)

Step 3: Subtract total straight flush hands from total flush hands

To find the number of flush hands that are not also straight flushes, we simply subtract the total combinations of straight flush hands from the total combinations of flush hands: \(\text{Flush hands (not straight flush)} = (4 \times \binom{13}{5}) - (4 \times 10)\) Finally, let's compute these results to get the answer:

Step 4: Compute the answer

Calculating the final result: \(\text{Flush hands (not straight flush)} = (4 \times \binom{13}{5}) - (4 \times 10)\) \(\text{Flush hands (not straight flush)} = (4 \times 1287) - (40)\) \(\text{Flush hands (not straight flush)} = 5148 - 40\) \(\text{Flush hands (not straight flush)} = 5108\) There are 5,108 different 5-card poker hands that consist of a flush, but not a straight flush.

Key Concepts

Combinatorics in Probability

Understanding the role of combinatorics in probability is fundamental when analyzing games of chance, such as poker. Combinatorics is the branch of mathematics dealing with combinations, permutations, and counts of various configurations that meet certain criteria. In probability, it helps us determine the number of possible outcomes in a situation.

To illustrate the concept using poker, imagine shuffling a standard deck of 52 cards. When dealing a 5-card hand, there are countless possible combinations. Combinatorics allows us to compute these possibilities using formulas like the combination formula, \( \binom{n}{k} \) which calculates the number of ways to choose \( k \) items from a set of \( n \) items without regard to the order.

Flush in Poker

A flush in poker is a hand where all five cards are of the same suit, but not in a sequential order. When exploring flush combinations, we must consider that suits in a deck of cards are distinct entities, and there are four of them: hearts, diamonds, clubs, and spades.

In calculating flush combinations, you select one of the four suits, and then choose five cards within that suit. Since order does not matter in this scenario, you use combinations, not permutations. The formula for the number of flush combinations \( (except for a straight flush) \) is then \( 4 \times \binom{13}{5} \) which calculates to 5,148 possible flush hands. However, to ensure straight flushes are excluded, we must subtract the number of straight flush combinations from this total.

Straight Flush in Poker

The straight flush represents a unique and powerful hand in poker. It incorporates both the concept of a straight and a flush; this hand consists of five cards in sequence, all of the same suit. It's simpler to calculate straight flush combinations than general flush combinations because the sequence element reduces the number of possibilities.

There are 10 possible sequences for each of the four suits, resulting in \( 4 \times 10 \) or 40 possible straight flush hands. This includes the elusive royal flush, which is the highest-ranking straight flush consisting of an ace, king, queen, jack, and ten of the same suit. When calculating flush hands, it's crucial to subtract these 40 straight flush combinations to avoid counting them twice.