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Magazine Chapter 5: Problem 65
A deck of playing cards contains 52 cards, four of which are aces. What is the probability that the deal of a five-card hand provides: a. A pair of aces? b. Exactly one ace? c. No aces? d. At least one ace?
Short Answer
Expert verified
Pair: 4.8%, One: 29.8%, None: 66.5%, At least one: 33.5%
Step by step solution
01
Understanding the Problem
Firstly, we need to understand that from a standard deck of 52 cards, we want to know the probability of specific combinations of aces in a five-card hand. There are 4 aces out of 52 cards in the deck.
02
Calculating Total Possible Hands
The total number of ways to choose a five-card hand from 52 cards is given by the combination formula \( \binom{52}{5} \). Calculate this to find the denominator for all probabilities related to these events.
03
Calculating Probability for a Pair of Aces
To find the probability of a pair of aces, we need to select 2 aces from 4 (\( \binom{4}{2} \)) and 3 non-aces from the remaining 48 cards (\( \binom{48}{3} \)). Multiply these results and divide by the total number of five-card hands, \( \binom{52}{5} \).
04
Calculating Probability for Exactly One Ace
For exactly one ace, choose 1 ace from 4 (\( \binom{4}{1} \)) and the other 4 cards from the 48 non-aces (\( \binom{48}{4} \)). Multiply and then divide by the total number of hands to find the probability.
05
Calculating Probability for No Aces
For no aces, all 5 cards must be chosen from the 48 non-aces (\( \binom{48}{5} \)). Calculate this and divide by the total number of hands to get the probability.
06
Calculating Probability for At Least One Ace
To find the probability of at least one ace, note that it is the complement of the event with no aces. So, subtract the probability of no aces from 1.
07
Probability Calculations
Finally, we calculate: - Total hands: \( \binom{52}{5} = 2,598,960 \).- Pair of aces: Probability \( = \frac{\binom{4}{2} \times \binom{48}{3}}{\binom{52}{5}} \).- Exactly one ace: Probability \( = \frac{\binom{4}{1} \times \binom{48}{4}}{\binom{52}{5}} \).- No aces: Probability \( = \frac{\binom{48}{5}}{\binom{52}{5}} \).- At least one ace: Probability \( = 1 - \frac{\binom{48}{5}}{\binom{52}{5}} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combination Formula
To understand card probability problems, it's crucial to master the combination formula. This formula helps us determine how many ways we can choose items from a larger set without regard to the order of selection. Mathematically, it's expressed as:
Where:
This formula enables us to calculate the total number of five-card hands when drawn from a 52-card deck using \( \binom{52}{5} \). It provides a basis for computing the probability of certain card combinations by serving as the denominator in our probability calculations.
- \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
Where:
- \( n \) is the total number of items to choose from (in our case, 52 cards).
- \( r \) is the number of items to select (like the 5 cards in our hand).
- \( ! \) denotes a factorial, meaning the product of all positive integers up to that number.
This formula enables us to calculate the total number of five-card hands when drawn from a 52-card deck using \( \binom{52}{5} \). It provides a basis for computing the probability of certain card combinations by serving as the denominator in our probability calculations.
Card Combinations
Card combinations examine the specific arrangement of cards in your hand. We look at different scenarios like having pairs, single cards, or categories like aces. To compute probabilities:
For example, to find the odds of having exactly one ace, you pick 1 ace from the 4 available and complete the hand with 4 more cards from the remaining 48 non-aces, calculated as \( \binom{4}{1} \times \binom{48}{4} \).
Understanding these combinations allows us to establish probabilities by relating them to the total possible hands.
- Pair of aces: Select 2 aces from 4, expressed as \( \binom{4}{2} \).
- Non-aces: Draw the rest of the hand with non-aces using \( \binom{48}{3} \) or \( \binom{48}{4} \) depending on the hand.
For example, to find the odds of having exactly one ace, you pick 1 ace from the 4 available and complete the hand with 4 more cards from the remaining 48 non-aces, calculated as \( \binom{4}{1} \times \binom{48}{4} \).
Understanding these combinations allows us to establish probabilities by relating them to the total possible hands.
Complement Rule For Probability
The complement rule is a powerful concept in probability that simplifies certain calculations. Instead of directly finding the probability of having at least one ace in your hand, it's often easier to use its complement, which is having no aces at all.
This approach is beneficial as follows:
Mathematically, it is expressed as:
This method can make your work smoother and is particularly handy when dealing with more complex scenarios. It prevents redundant calculations by focusing on a simpler alternative.
This approach is beneficial as follows:
- Calculate the probability of drawing no aces using \( \binom{48}{5} \).
- Subtract this probability from 1 to find the complement: the probability of having at least one ace.
Mathematically, it is expressed as:
- \( P( ext{at least one ace}) = 1 - P( ext{no aces}) \)
This method can make your work smoother and is particularly handy when dealing with more complex scenarios. It prevents redundant calculations by focusing on a simpler alternative.
