Problem 22 - 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. An ace, king, queen, jack, and ten of the same suit (royal flush)

Step-by-Step Solution

Answer

The probability of a royal flush is \( \frac{1}{649,740} \).

Step 1: Understanding the Problem

We need to find the probability of being dealt a royal flush in poker. A royal flush consists of the ace, king, queen, jack, and ten, all of the same suit.

Step 2: Counting Favorable Outcomes

There are four suits in a deck of cards (hearts, diamonds, clubs, spades). A royal flush can be composed of any of these suits, meaning there are 4 possible royal flushes in a deck.

Step 3: Counting Possible Outcomes

The total number of ways to deal 5 cards from a 52-card deck is given by the combination formula: \( \binom{52}{5} \). Calculating this gives us \( 2,598,960 \) possible 5-card hands.

Step 4: Calculating the Probability

The probability of drawing a royal flush is the number of favorable outcomes divided by the total number of outcomes. Thus, the probability is \( \frac{4}{2,598,960} \).

Step 5: Simplifying the Probability

Simplifying the fraction \( \frac{4}{2,598,960} \), we get \( \frac{1}{649,740} \). This is the probability of being dealt a royal flush.

Key Concepts

Combinatorics

In mathematics, combinatorics is the study of counting, arranging, and combination of elements within a set according to certain constraints. It often involves evaluating the total number of ways certain objects can be selected or arranged. Combinatorics is key to understanding complex probability problems, such as calculating the likelihood of poker hands.

One of the fundamental concepts in combinatorics is the "combination," which represents how many different ways you can choose a subset of objects from a larger set without regard to the order of selection. The formula for combinations is given by:

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

where \( n \) is the total number of items to choose from, and \( r \) is how many items are to be chosen.

For example, when calculating the number of possible 5-card hands from a 52-card deck, we use the combination formula to find:

• \(\binom{52}{5} = 2,598,960\)

This tells us there are 2,598,960 different possible 5-card poker hands.

Poker Hands

Poker is a popular card game that involves betting and individual play. The game often uses a standard deck of 52 cards, and understanding poker hands is crucial to strategize effectively.

A poker hand can consist of specific combinations of cards, with each combination having various ranks based on rarity and difficulty to attain. Some examples of poker hands include:

• High Card - the simplest hand, where no other combinations are made.
• One Pair - any two cards of the same rank.
• Full House - three cards of one rank and two of another.
• Straight Flush - five consecutive cards of the same suit.

Among the rarest and highest-ranked poker hands is the "royal flush," which ranks even higher than the straight flush and involves an ace, king, queen, jack, and ten, all of the same suit. This brings us to our next concept, the royal flush.

Royal Flush

A royal flush is the most prestigious hand in poker, often considered unbeatable in poker games that use a standard deck. It consists of a sequence of the highest-ranking cards: ace, king, queen, jack, and ten, all within the same suit.

The rarity of a royal flush makes it particularly special. In a 52-card deck, there are four suits (hearts, diamonds, clubs, spades), meaning there are exactly four possible royal flush combinations.

Calculating the probability of drawing a royal flush involves determining the ratio of favorable outcomes to all possible outcomes:

• Favorable Outcomes: 4 (one for each suit).
• Total Possible 5-card Hands: 2,598,960.
• Probability: \( \frac{4}{2,598,960} \)
• After simplification: \( \frac{1}{649,740} \)

This translates to a tiny probability of roughly 0.00015%, illustrating the difficulty and excitement associated with drawing a royal flush.