Problem 24 - Algebra for College Students

Question

\text { A king is drawn. } \frac{1}{13}

Step-by-Step Solution

Answer

The probability of drawing a king is \( \frac{1}{13} \).

Step 1: Understanding the Problem

We need to calculate the probability of drawing a king from a standard deck of 52 playing cards. There are 4 suits, each with 13 cards, and each suit contains one king.

Step 2: Count the Total Number of Possible Outcomes

Since we are drawing from a deck of 52 cards, there are 52 possible outcomes.

Step 3: Count the Number of Favorable Outcomes

The favorable outcomes are the cases where a king is drawn. There is 1 king in each of the 4 suits, making a total of 4 kings.

Step 4: Calculate the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability of drawing a king is calculated as \( \frac{4}{52} \).

Step 5: Simplify the Fraction

The fraction \( \frac{4}{52} \) can be simplified by dividing the numerator and the denominator by 4, giving \( \frac{1}{13} \). This simplified fraction represents the probability of drawing a king.

Key Concepts

Standard Deck of Cards

A standard deck of cards is a familiar tool in probability exercises. A deck consists of 52 cards, which are divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, ranging from Ace through King. This structured setup allows us to calculate probabilities based on known possibilities.

When working with probability problems involving a deck of cards, the total number of possible outcomes is typically 52, considering each card could potentially be the outcome.

• The four suits are of equal size, each containing 13 cards.
• Each suit follows a traditional rank order, such as Ace, 2, 3, ..., through King.
• Understanding the uniform structure of the deck is essential for correctly setting up and solving probability scenarios.

This consistency makes a deck of cards an excellent model for exploring basic probability questions, such as the likelihood of drawing a specific card.

Favorable Outcomes

In probability, understanding favorable outcomes is crucial. A favorable outcome is simply the event you are interested in.

In the problem about drawing a king, our favorable outcomes are drawing one of the four kings from the deck. Since there are four suits and each suit contains exactly one king:

• The number of favorable outcomes for drawing a king is four.
• Each favorable outcome represents drawing a king from one of the four suits.

To calculate the probability of an event, it's important to determine how many ways the event can occur. This understanding allows you to divide the number of favorable outcomes by the total number of outcomes. Clearly identifying and understanding these favorable outcomes effectively sets you up for accurate probability calculations.

Simplifying Fractions

Simplifying fractions in probability is a helpful step that makes understanding the likelihood of an event easier and cleaner.

When calculating probabilities, the result is often a fraction that represents the ratio of favorable outcomes to the total number of possible outcomes. In the case of drawing a king, the probability started as \( \frac{4}{52} \) because there are 4 kings in a deck of 52 cards.

• Simplifying means reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• In \( \frac{4}{52} \), both 4 and 52 can be divided by 4, the GCD, resulting in \( \frac{1}{13} \).

Simplifying fractions like \( \frac{4}{52} \) to \( \frac{1}{13} \) not only makes computations clearer, but it often produces a more manageable number to work with, without changing the inherent likelihood it represents.