Problem 23 - Algebra for College Students

Question

One card is drawn from a standard deck of 52 playing cards. Find the probability of each of the following events: \text { A heart is drawn. } \frac{1}{4}

Step-by-Step Solution

Answer

The probability of drawing a heart is \( \frac{1}{4} \).

Step 1: Understanding the Problem

We need to find the probability of drawing a heart from a standard deck of 52 playing cards.

Step 2: Identifying the Deck and the Specific Event

A standard deck of cards contains 52 cards, divided equally among four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards. We need to determine the probability of drawing one of the 13 hearts.

Step 3: Using the Probability Formula

The probability of an event is given by the formula \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). Here, the number of favorable outcomes is 13 (hearts), and the total possible outcomes are 52 (all cards in the deck).

Step 4: Calculating the Probability

Using the probability formula, we find the probability of drawing a heart: \( P( ext{Heart}) = \frac{13}{52} = \frac{1}{4} \).

Step 5: Checking the Calculation

To ensure the accuracy of our calculation, simplify \( \frac{13}{52} \) by dividing both the numerator and the denominator by 13 to get \( \frac{1}{4} \). The calculation is correct.

Key Concepts

Standard Deck of Cards

A standard deck of cards is universally used in many card games worldwide. It consists of 52 cards, which adds structure to the variety of games played. All 52 cards are divided equally into four suits: **hearts**, **diamonds**, **clubs**, and **spades**. Each suit contains 13 cards, ranging from Ace through King. This uniform composition not only brings order but also creates a foundation for understanding basic probability in card games. Whether you're a casual player or a math enthusiast, comprehending a standard deck is essential for analyzing and predicting outcomes in card games.

Favorable Outcomes

In probability, "favorable outcomes" refer to the outcomes that fulfill the condition we are considering. For example, if we're trying to determine the chance of drawing a heart from a standard deck, all the hearts in the deck are considered favorable outcomes.

• The number of favorable outcomes of drawing a heart is 13 because there are 13 hearts in the deck.
• Understanding what constitutes a favorable outcome helps us accurately apply the probability formula.

Identifying these outcomes correctly is crucial for solving probability problems and predicting events in a structured way.

Probability Formula

The probability of an occurrence can be calculated with a straightforward formula. This formula is often expressed as:\[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]Here:

• **P(E)** represents the probability of the event occurring.
• The **favorable outcomes** are the successful scenarios that satisfy the event condition.
• The **total possible outcomes** are the number of all possible events that can occur. For a standard deck, this number is 52.

Using this formula enables us to calculate probabilities easily and understand how likely certain outcomes are in any given situation.

Suits in Cards

The concept of suits in a standard deck can be fascinating and vital for learning card probability.
Each suit—hearts, diamonds, clubs, and spades—consists of 13 cards, making a total of 52 cards across all four suits. Understanding suits helps clarify which cards can satisfy specific event conditions, such as drawing a heart. This organized structure:

• Gives a systematic approach to playing and strategizing in card games.
• Allows clear calculations of probabilities for different events involving specific suits.

Recognizing how suits operate within the deck is a fundamental aspect of both gameplay and mathematical calculations involving cards.