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Chapter 5: Problem 11

Drawing a Card Suppose that a single card is selected from a standard 52 -card deck. What is the probability that the card drawn is a club? Now suppose that a single card is drawn from a standard 52 -card deck, but we are told that the card is black. What is the probability that the card drawn is a club?

Short Answer

Expert verified
The probability of drawing a club is \(\frac{1}{4}\). Given the card is black, the probability it is a club is \(\frac{1}{2}\).

Step by step solution

01

Understand the Deck Composition

A standard 52-card deck consists of 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards.
02

Calculate the Total Number of Cards

There are a total of 52 cards in the deck. This is important for calculating probabilities.
03

Find the Number of Clubs

Since there are 4 suits and each suit has 13 cards, the number of clubs is 13.
04

Calculate the Probability of Drawing a Club

The probability of drawing a club from the deck is the number of clubs divided by the total number of cards. Therefore, the probability is \(\frac{13}{52}\).
05

Simplify the Probability

Simplify the fraction \(\frac{13}{52}\) to get \(\frac{1}{4}\). This means the probability is 1/4 or 25%.
06

Understand the Condition of Drawing a Black Card

There are only two black suits in a standard deck: clubs and spades. Each suit has 13 cards. So, there are 26 black cards in total.
07

Calculate the Probability Given the Card is Black

Given that the card is black (26 cards), and we want the card to be a club (13 clubs), the probability is \(\frac{13}{26}\).
08

Simplify the Conditional Probability

Simplify the fraction \(\frac{13}{26}\) to get \(\frac{1}{2}\). This means the probability is 1/2 or 50%.

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Key Concepts

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

Understanding the Deck of Cards
To understand probability in card games, you first need to be familiar with the composition of a standard deck of cards. A standard deck contains 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards. By knowing this, you can start computing probabilities for different outcomes when drawing cards from the deck.

For example, each suit (hearts, diamonds, clubs, spades) has 13 cards, and knowing this helps in calculating probabilities for drawing a specific suit.
Calculating Suit Probability
The probability of drawing a specific suit, such as clubs, from a standard 52-card deck can be calculated using basic probability principles. The number of clubs in the deck is 13 out of 52 cards. To find this probability, you use the formula:
\( P(A) = \frac{\text{Number of Clubs}}{\text{Total Number of Cards}} = \frac{13}{52} \).

When simplified, \( \frac{13}{52} = \frac{1}{4} \), equating to a 25% chance of drawing a club. This simplification process is important to make probabilities more intuitive.
Understanding Conditional Probability
Conditional probability refers to finding the probability of an event happening given that another event has already occurred. In this case, we need to find the probability of drawing a club given that we know the card is black.

Out of the 52 cards, 26 are black (clubs and spades). Given that the card is already known to be black, we only focus on these 26 cards. Since there are 13 clubs in these 26 black cards, the conditional probability is:
\( P(A|B) = \frac{\text{Number of Clubs}}{\text{Total Number of Black Cards}} = \frac{13}{26} \).

Simplifying \( \frac{13}{26} = \frac{1}{2} \), the probability is 50%, meaning there's a 50% chance a black card drawn is a club.
Simplifying Fractions
Simplification is a crucial step in probability calculations to make answers more understandable. When you have a fraction like \( \frac{13}{52} \) for the probability of drawing a club, you simplify it by dividing the numerator and the denominator by their greatest common divisor, which is 13 in this case.

Simplifying \( \frac{13}{52} \) to \( \frac{1}{4} \) makes it easier to interpret as a 25% chance. Similarly, \( \frac{13}{26} \) simplified to \( \frac{1}{2} \) clearly shows there's a 50% chance a black card drawn will be a club. Using simplified fractions helps in grasping the core idea of probabilities more effectively.

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