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Chapter 12: Problem 16

Five cards are chosen at random from a deck of 52 cards. Find the probability that the set contains a) 3 kings b) 4 hearts and 1 diamond.

Short Answer

Expert verified
a) Probability is approximately 0.0017; b) Probability is 0.00495.

Step by step solution

01

Understanding the Problem

We need to find the probability of specific card combinations from a standard deck of 52 cards. We'll tackle two separate situations: drawing 3 kings and the rest any cards (except for being all four kings) and drawing 4 hearts with 1 diamond.
02

Calculate Total Possible Outcomes

For both situations, the total number of ways to choose any 5 cards from 52 is needed. This can be found using combinations: \( \binom{52}{5} \).
03

Finding Ways to Draw 3 Kings

First, find out how many ways we can choose 3 kings from 4 available in the deck: \( \binom{4}{3} \). Then, choose the remaining 2 cards from the 48 non-king cards: \( \binom{48}{2} \).
04

Calculate Probability for 3 Kings

Multiply the two numbers from Step 3 and divide by the total outcomes from Step 2 to find the probability for drawing 3 kings: \( \frac{\binom{4}{3} \times \binom{48}{2}}{\binom{52}{5}} \). Simplify this to get the final probability.
05

Finding Ways to Draw 4 Hearts and 1 Diamond

Determine the number of ways to choose 4 hearts from the 13 available: \( \binom{13}{4} \). Then, choose 1 diamond from the 13 available diamonds: \( \binom{13}{1} \).
06

Calculate Probability for 4 Hearts and 1 Diamond

Multiply the results from Step 5 and divide by the total number of outcomes from Step 2 to get the probability of drawing 4 hearts and 1 diamond: \( \frac{\binom{13}{4} \times \binom{13}{1}}{\binom{52}{5}} \). Simplify this to find the final answer.

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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 deals with counting, arranging, and combining objects. It provides tools to solve problems related to the arrangement of items and determination of various groupings. A common tool within combinatorics used in the above exercise is the combination formula which is represented by the symbol \( \binom{n}{k} \). This formula is used to find the number of ways to choose \( k \) items from \( n \) items without regard to the order of selection.
For example:
  • Choosing 5 cards from a deck of 52 involves calculating \( \binom{52}{5} \).
  • Choosing 3 kings from 4 available in the deck involves \( \binom{4}{3} \).
Combinatorics helps make complex problems more manageable by breaking them down into combinations and applying systematic formulas.
Card Probability
The concept of card probability involves calculating the likelihood of specific outcomes from a standard deck of cards. A deck consists of 52 cards split into four suits (hearts, diamonds, clubs, and spades) with thirteen ranks (Ace to King).
To calculate probabilities with cards, you need to define different events and use combinations to find the favorable outcomes over total possible outcomes. Here's how it works in the exercise:
  • For finding the probability of drawing 3 kings, you calculate the number of ways to choose 3 kings and combine it with the number of ways to choose the remaining cards, then divide by the total combinations of 5 cards from the deck.
  • For 4 hearts and 1 diamond, determine the number of combinations to select the hearts and one diamond, then proceed similarly to divide by the total possible outcomes.
This process relies on understanding the makeup and rules of a deck and utilizes combinatorial principles to determine probabilities.
Discrete Mathematics
Discrete mathematics is a field of study focused on mathematical structures that are discrete rather than continuous. This includes areas such as graph theory, logic, and combinatorics. It is particularly useful in computer science and any domain requiring finite or countable structures. In the context of this exercise, discrete mathematics comes into play through:
  • Using combinatorial calculations to assess distinct card arrangements from the deck, which is a discrete structure.
  • Applying rules of discrete probability where the sample space (all possible hands of cards) is finite, allowing for precise calculation of probabilities.
With its focus on precise, separate values, discrete mathematics provides the foundational principles needed to solve the problem of card probabilities accurately. It allows tackling questions that involve definite, countable outcomes, as seen in the problem of drawing specific kinds of cards from a deck.

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Most popular questions from this chapter

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