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Magazine Chapter 12: Problem 49
Five cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of exactly two pairs?
Short Answer
Expert verified
The probability of drawing exactly two pairs is approximately 0.0475.
Step by step solution
01
Understand the Problem
We need to calculate the probability of drawing exactly two pairs from a standard deck of 52 cards when choosing 5 cards without replacement. "Two pairs" means two sets of cards of the same rank and one card of a different rank.
02
Determine Total Possible Outcomes
The total number of ways to choose any 5 cards from a deck of 52 is given by the combination formula \( \binom{52}{5} \). Calculating this gives: \( \binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960 \).
03
Calculate Favorable Outcomes for Two Pairs
To form two pairs, choose 2 distinct ranks out of 13 possible. First, select 2 ranks for the pairs. This is \( \binom{13}{2} \). For each rank, choose 2 out of 4 cards, \( \binom{4}{2} \) for each rank. The fifth card will come from the remaining 11 ranks, and for the single card, choose 1 rank from these remaining ranks and 1 card from the 4 available cards: \( \binom{11}{1} \times \binom{4}{1} \). Thus, the total number of favorable ways is \( \binom{13}{2} \times \binom{4}{2}^2 \times \binom{11}{1} \times \binom{4}{1} \). Calculating these gives: \( 78 \times 6 \times 6 \times 44 = 123,552 \).
04
Calculate the Probability
The probability is the ratio of favorable outcomes to the total possible outcomes: \( P(\text{two pairs}) = \frac{123,552}{2,598,960} \approx 0.0475 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combinatorics
Combinatorics is a fundamental concept in mathematics that deals with counting, arrangement, and combination of elements from a set. It's a key part of probability calculations. When solving problems like the one involving drawing five cards and finding specific patterns, we often use combination formulas.
A combination is a method of selecting items from a larger group where the order does not matter. This is different from permutations where the order is important. In the card problem, combinations are used because when you draw cards, the order doesn't matter; only the selection does.
To use the combination formula, you calculate \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. For example, choosing 5 cards from 52 cards is expressed as \( \binom{52}{5} \). The formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In the exercise, this helps us determine the total possible outcomes as well as the favorable outcomes. This understanding of combinations is crucial for solving similar probability problems.
A combination is a method of selecting items from a larger group where the order does not matter. This is different from permutations where the order is important. In the card problem, combinations are used because when you draw cards, the order doesn't matter; only the selection does.
To use the combination formula, you calculate \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. For example, choosing 5 cards from 52 cards is expressed as \( \binom{52}{5} \). The formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In the exercise, this helps us determine the total possible outcomes as well as the favorable outcomes. This understanding of combinations is crucial for solving similar probability problems.
Card Probability
Card probability refers to the likelihood of drawing a specific hand or combination of cards from a standard deck. A standard deck consists of 52 cards and is divided into four suits, each with 13 ranks: hearts, diamonds, clubs, and spades.
When calculating the probability of drawing cards, it's crucial to understand what a 'hand' means. A 'hand' is a specific selection, such as pairs, flushes, or full houses. In our exercise, we focused on finding the probability of drawing exactly two pairs, which is two sets of cards of the same rank and one card of a different rank.
To find the probability, we calculate two main components:
When calculating the probability of drawing cards, it's crucial to understand what a 'hand' means. A 'hand' is a specific selection, such as pairs, flushes, or full houses. In our exercise, we focused on finding the probability of drawing exactly two pairs, which is two sets of cards of the same rank and one card of a different rank.
To find the probability, we calculate two main components:
- Total possible outcomes, using combinations, to know how many ways we can draw a certain number of cards.
- Favorable outcomes, or the number of ways to achieve the specific hand, again using combinations.
Drawing Without Replacement
Drawing without replacement describes a scenario where once an item is drawn, it cannot be placed back into the original set before the next item is drawn. This concept is important because it affects how we calculate probability.
In our card problem, since the cards are drawn without replacement, the deck size reduces with each draw. This means that the number of possible outcomes changes with each successive draw. For example, drawing the second card is dependent on the first card still being out of the deck, thus making probability calculations dynamic.
This affects both the calculations of total possible outcomes and the specific sequences we consider as favorable outcomes. For example, when calculating the favorable outcomes for two pairs, we adjust combinations to account for previous cards that have already been drawn.
Understanding the concept of drawing without replacement is crucial for accurately computing probabilities in situations like card games where replacement isn't an option.
In our card problem, since the cards are drawn without replacement, the deck size reduces with each draw. This means that the number of possible outcomes changes with each successive draw. For example, drawing the second card is dependent on the first card still being out of the deck, thus making probability calculations dynamic.
This affects both the calculations of total possible outcomes and the specific sequences we consider as favorable outcomes. For example, when calculating the favorable outcomes for two pairs, we adjust combinations to account for previous cards that have already been drawn.
Understanding the concept of drawing without replacement is crucial for accurately computing probabilities in situations like card games where replacement isn't an option.
