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Chapter 4: Problem 8

Find the probability of getting a full house \((3\) cards of one denomination and 2 of another) when 5 cards are dealt from an ordinary deck.

Short Answer

Expert verified
The probability of getting a full house when 5 cards are dealt is approximately 0.00144.

Step by step solution

01

Understanding a Full House

A full house consists of three cards of one rank (e.g., three 7s) and two cards of another rank (e.g., two Kings). A standard deck has 52 cards, divided into 4 suits.
02

Choosing the Rank for Three Cards

There are 13 possible ranks in a deck. To choose the rank for the three cards, we have \( \binom{13}{1} \) ways, which is 13.
03

Selecting Suits for Three Cards

Once a rank is chosen for the three cards, there are 4 suits, and we need to choose 3 out of these 4. The number of ways to choose 3 suits from 4 is \( \binom{4}{3} = 4 \).
04

Choosing the Rank for Two Cards

For the pair of two cards, we must choose a different rank from the 12 remaining ranks. We can choose this rank in \( \binom{12}{1} \) ways, which is 12.
05

Selecting Suits for Two Cards

After choosing the rank for the two cards, similar to Step 3, there are 4 suits, and we need to choose 2. The number of ways to choose 2 suits from 4 is \( \binom{4}{2} = 6 \).
06

Calculating Total Full House Combinations

The total number of full house combinations is the product of the number of combinations from Steps 2 to 5: \( 13 \times 4 \times 12 \times 6 = 3,744 \).
07

Calculating Total 5-Card Hand Combinations

The total number of possible 5-card hands from a deck of 52 cards is \( \binom{52}{5} = 2,598,960 \).
08

Finding the Probability of a Full House

The probability of being dealt a full house is the number of full house combinations divided by the total number of 5-card hands: \( \frac{3,744}{2,598,960} \approx 0.00144 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combination of objects. It plays a crucial role in calculating probabilities in card games, like the odds of getting a full house when dealt 5 cards from a standard deck.
  • It involves determining different possible ways things can happen.
  • With combinatorics, we can calculate the number of ways to pick card ranks and suits from a deck.
A key aspect of combinatorics is the binomial coefficient, often represented as \( \binom{n}{k} \), which indicates the number of ways to choose \( k \) items from \( n \) without regard to order. For example, choosing 3 suits from 4 possible ones is calculated as \( \binom{4}{3} = 4 \). This formula is vital when solving card probability problems.
Card Games
Card games are popular recreational activities that often require a blend of strategy and luck. A common aspect of these games is understanding the probability involved, which can significantly influence strategic decisions.
  • In games like poker, understanding hand probabilities helps make informed decisions.
  • Knowledge of card combinations aids in predicting opponents' hands.
Each game has its rules that determine how cards can be combined to form different hands. For example, in poker, a full house is a valuable hand comprising three cards of one rank and two of another.
Full House Probability
Calculating the probability of obtaining a full house in a 5-card hand involves several steps. This involves using combinatorics to determine both the number of favorable outcomes and the total possible outcomes.
  • A full house is made by choosing one rank for three cards and another rank for two cards.
  • The number of possible full house hands is calculated as a product of selecting ranks and choosing suits for those ranks.
For instance, we calculate 13 ways to choose the three-card rank, 4 ways to choose the suits for these cards, 12 ways to choose the two-card rank, and 6 ways to choose their suits, totaling 3,744 full house combinations. This probability is the quotient of this product and the total possible 5-card hands, approximately 0.00144, or 0.144%.
Deck of Cards
A standard deck of cards is a collection of 52 playing cards used in various games. It comprises four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 ranks: Ace, 2 through 10, Jack, Queen, and King.
  • All suits are equally likely to be drawn, making it easy to calculate odds.
  • The uniform structure of decks is instrumental in probabilities and combinatorial calculations.
The identical number of ranks and suits for each, regardless of the suit, simplifies mathematical operations. Understanding the composition of a deck is fundamental in calculating probabilities like those in card games, for instance, assessing how likely one is to get a full house from a deck.

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Most popular questions from this chapter

First-year students at a particular college must take one English class, one class in mathematics, a first-year seminar, and an elective. There are 2 English classes to choose from, 3 mathematics classes, 5 electives, and everyone takes the same first-year seminar. Represent the possible schedules, using a tree diagram.

A medical doctor stated that with a certain treatment, a patient has a \(50 \%\) chance of recovering without surgery. That is, "Either he will get well or he won't get well." Comment on this statement.

Wake Up cereal comes in 2 types, crispy and crunchy. If a researcher has 10 boxes of each, how many ways can she select 3 boxes of each for a quality control test?

The state narcotics bureau must form a 5-member investigative team. If it has 25 agents from which to choose, how many different possible teams can be formed?

How many ways can a dinner patron select 3 appetizers and 2 vegetables if there are 6 appetizers and 5 vegetables on the menu?
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