Problem 21 - Algebra and Trigonometry

Question

\(19-22\) . A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. Five face cards

Step-by-Step Solution

Answer

The probability is approximately 0.000305.

Step 1: Understand Face Cards

In a standard deck of 52 cards, there are three face cards in each of the four suits: Kings, Queens, Jacks. Therefore, there are 3 face cards per suit times 4 suits, which equals 12 face cards total.

Step 2: Compute Total Possible Hands

A poker hand consists of 5 cards. Thus, the total number of possible 5-card hands from a 52-card deck is given by the combination formula \( \binom{52}{5} = 2,598,960 \). This represents all possible 5-card hands that can be made from the deck.

Step 3: Compute Number of All-Face Card Hands

We need to select 5 face cards from the 12 available face cards. The number of ways to select 5 face cards from 12 is calculated using the combination formula: \( \binom{12}{5} = 792 \). This represents the number of hands consisting entirely of face cards.

Step 4: Calculate the Probability

The probability is the number of favorable outcomes (all-face card hands) divided by the total number of possible outcomes (all 5-card hands). Thus, the probability is given by the formula: \[ P(\text{all face cards}) = \frac{\binom{12}{5}}{\binom{52}{5}} = \frac{792}{2,598,960} \approx 0.000305 \].

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. In the context of our poker problem, it helps us determine how many ways we can choose certain cards. Combinations are particular arrangements where the order does not matter.

To calculate combinations, we use the combination formula, which is expressed as:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \(n\) is the total number of items to choose from, \(r\) is the number of items to be chosen, and \(!\) is the factorial operation, which multiplies a series of descending natural numbers. For example, to choose 5 cards from 52, we calculate \( \binom{52}{5} \).

Combinatorics is essential in probability because it helps us count the number of possible outcomes, which is crucial for determining probabilities in scenarios like poker hands.

Poker Hand Probability

Poker hand probability refers to the likelihood of getting a specific type of hand when dealing a standard 5-card hand from a 52-card deck. The probability helps players understand their chances of receiving certain hands, such as flushes, straights, or face card hands like in the exercise.

The probability of an event happening is calculated using the formula:

• Probability = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

In this exercise, the number of favorable outcomes is the number of ways to draw five face cards, calculated as \( \binom{12}{5} \), and the total number of possible hands is \( \binom{52}{5} \).

Understanding poker hand probabilities can significantly impact strategic game decisions. These calculations allow players to assess risk and possible reward, enhancing gameplay strategy.

Face Cards

Face cards in a standard deck of 52 cards include the Kings, Queens, and Jacks. Each suit (Hearts, Diamonds, Clubs, Spades) contains one of each face card, making a total of 12 face cards in the deck.

Knowing the composition of face cards is crucial in calculating probabilities for games like poker. In our exercise, the face cards are the key to solving the probability problem. We specifically aimed to find a hand containing only these face cards.

• There are 3 face cards in each of the 4 suits.
• This results in 12 face cards in total (3 face cards/suit × 4 suits).

With only 12 to choose from, finding a hand of all face cards is less probable than hands with a mixture of card types, demonstrating why combinations and understanding card types are important in probability theory.