Problem 57 - College Algebra with Modeling and Visualization

Question

\(\mathbf{A}\) card is drawn from a standard deck of 52 cards. Given that the card is a face card, what is the probability that the card is a king? (Hint; A face card is a jack, queen, or king.)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{3} \).

Step 1: Define the Sample Space

To solve this problem, we first need to establish our sample space. Given that the card is a face card, we are dealing with the subset of face cards from a standard deck. A standard deck has 52 cards, and among these, there are 3 face cards in each suit: a Jack, Queen, and King. Hence, the total number of face cards is 4 suits multiplied by 3 face cards per suit, which is 12.

Step 2: Identify Successful Outcomes

Next, we identify the successful outcomes for our problem. We specifically want to know the probability that the face card drawn is a King. There is one King in each of the four suits (hearts, spades, diamonds, clubs), making a total of 4 Kings in the deck.

Step 3: Compute the Probability

Using the definition of probability, which is the number of successful outcomes divided by the total number of possible outcomes, we can compute the probability of drawing a King given a face card. Successful outcomes (Kings) = 4, Total face cards = 12, therefore the probability is \( \frac{4}{12} = \frac{1}{3} \).

Key Concepts

Sample Space in Conditional Probability

When dealing with conditional probability, understanding the concept of a **sample space** is crucial. A sample space is simply the set of all possible outcomes of a certain experiment or scenario. For example, if you flip a coin, the sample space consists of two outcomes: heads and tails. In the context of our card problem, where we draw from a standard deck of cards, we first define a sample space that suits our condition. Given the situation where the card drawn is a face card, we create a subset of the possible outcomes. In a standard 52-card deck, there are 12 face cards (3 face cards per each of the 4 suits). Therefore, the sample space considering only face cards contains 12 possible outcomes:

• King of Hearts, Queen of Hearts, Jack of Hearts
• King of Spades, Queen of Spades, Jack of Spades
• King of Diamonds, Queen of Diamonds, Jack of Diamonds
• King of Clubs, Queen of Clubs, Jack of Clubs

This sample space helps us focus only on the relevant possibilities, making it easier to calculate probabilities related to our specific condition.

Understanding Probability

**Probability** is a fundamental concept in statistics and mathematics, used to quantify the likelihood of an event occurring. To put it simply, probability measures how likely an event is to happen. It ranges from 0 to 1, where 0 means that an event will not occur at all, and 1 is certain to occur.To calculate probability, you use the formula: \[P( ext{Event}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}}\]In our card problem, the total number of possible outcomes is determined by the sample space of face cards. Among the 12 face cards, we want to find the probability that a drawn card is a King. Since there are 4 Kings, the probability is calculated by dividing the number of Kings by the total number of face cards:\[P( ext{King}) = \frac{4}{12} = \frac{1}{3}\]It's important to recognize that probability is not just about luck but a mathematical expression of likelihood based on a given set of conditions.

Exploring a Standard Deck of Cards

Understanding a **deck of cards** is essential when tackling probability problems involving cards. A standard deck contains 52 cards, divided into four suits: hearts, spades, diamonds, and clubs, each consisting of 13 cards. Each suit includes:

• Number cards from 2 to 10
• Three face cards: Jack, Queen, King
• Ace

Face cards are especially intriguing because, in many games and probability problems, they are often singled out due to their unique properties and grouping. In our exercise, the key focus was on face cards, comprising the Jack, Queen, and King of each suit. Within the deck's context here, the selection of 12 face cards already filters the deck down to a selective group, simplifying our conditional probability problem. Knowing how decks are structured helps solve broader card-related problems. Whether you're playing a game or solving a probability exercise, being familiar with how a deck is split and counted significantly impacts your approach and understanding of such problems.