Problem 26 - College Algebra

Question

For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: A club

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to find the probability of drawing a club from a standard deck of 52 cards. A standard deck includes four suits: hearts, diamonds, clubs, and spades, each containing 13 cards.

Step 2: Identifying Favorable Outcomes

We identify the number of favorable outcomes. Since there are 13 clubs in a deck, the number of favorable outcomes is 13.

Step 3: Identifying Total Outcomes

Next, we identify the total number of possible outcomes when drawing one card. Since there are 52 cards in the deck, the total number of outcomes is 52.

Step 4: Calculate the Probability

Using the probability formula, \( P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \), we find the probability of drawing a club: \( P( ext{a club}) = \frac{13}{52} \).

Step 5: Simplify the Probability

Simplify the fraction \( \frac{13}{52} \) by dividing both the numerator and the denominator by 13. This gives \( \frac{1}{4} \). Therefore, the probability of drawing a club is \( \frac{1}{4} \).

Key Concepts

Standard Deck of Cards

A standard deck of cards is a common tool used in probability and statistics. It consists of 52 cards divided equally among four suits: hearts, diamonds, clubs, and spades.
Each suit contains 13 cards, ranging from 2 to 10 and including a jack, queen, king, and an ace.
Understanding the composition of a standard deck is essential. It helps when calculating probabilities, especially when you're learning basic probability concepts.

• Hearts and diamonds are red suits.
• Clubs and spades are black suits.

The even distribution of suits and values plays a significant role in probability exercises and experiments.

Favorable Outcomes

In probability, a favorable outcome is simply the result we are interested in. For instance, when calculating the probability of drawing a club, the favorable outcomes are the cards that are clubs.
In our given problem, there are 13 clubs in a standard deck of cards. Thus, there are 13 favorable outcomes.
Understanding what constitutes favorable outcomes is crucial because it forms the basis of calculating any probability.

Probability Formula

The probability formula is a fundamental concept used to determine the likelihood of a specific event happening. The formula is written as:
\[ P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \] Where:

• \( P(E) \) represents the probability of the event \( E \) occurring.
• The numerator, "Number of Favorable Outcomes," is the count of desired results.
• The denominator, "Total Number of Outcomes," is the overall number of possible results.

In our card problem, the probability of drawing a club is calculated by dividing the 13 club cards by the 52 total cards.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It makes probabilities easier to interpret and understand. When you simplify a fraction, you reduce it to its smallest possible form by dividing both the numerator and the denominator by their greatest common divisor.For instance, in the card probability exercise, the fraction \( \frac{13}{52} \) is simplified by dividing by 13, resulting in \( \frac{1}{4} \).
This process makes it clear that there is a 1 in 4 chance, or 25% probability, of drawing a club from the deck.