Problem 26 - College Algebra with Corequisite Support

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 calculate the probability of drawing a club from a standard deck of 52 cards.

Step 2: Identify Total Outcomes

A standard deck of cards contains 52 cards. These 52 cards represent all possible outcomes when drawing one card from the deck.

Step 3: Identify Favorable Outcomes

In a deck of cards, there are 4 suits: hearts, diamonds, spades, and clubs. Each suit has 13 cards. Therefore, the favorable outcomes, i.e., drawing a club, are 13 cards.

Step 4: Apply Probability Formula

Probability is calculated using the formula: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Here, the number of favorable outcomes is 13 (clubs), and the total number of outcomes is 52.

Step 5: Calculate the Probability

Substitute the numbers into the formula: \( P(\text{Club}) = \frac{13}{52} \). Simplify this fraction to \( \frac{1}{4} \).

Step 6: Conclude the Solution

The probability of drawing a club from a standard deck of 52 cards is \( \frac{1}{4} \). This means there is a 25% chance to draw a club.

Key Concepts

Standard Deck of Cards

A standard deck of cards is a common tool in probability exercises. It consists of 52 cards, divided into four suits: hearts, diamonds, spades, and clubs. Each suit contains 13 cards, ranging from Ace to King.

• Each card in a suit has the same rank in all the other suits, meaning there are 13 ranks per suit.
• The suits are divided into two colors: red (hearts and diamonds) and black (spades and clubs).
• This organized structure of the deck makes it easier to perform probability calculations, as the total number of outcomes is always 52.

Knowing these basic components of a standard deck helps us to swiftly understand outcomes and analyze probability in card-drawing scenarios.

Favorable Outcomes

In probability, "favorable outcomes" imply the outcomes we are interested in observing or calculating. Specifically, when dealing with a card-drawing exercise, favorable outcomes align with the question posed, such as drawing a specific suit or card. Let's consider the example: Calculating the probability of drawing a club.

• Since there are four suits, each with 13 cards, the favorable outcomes for drawing a club are precisely 13 cards (Ace through King of clubs).
• This relevance simplifies part of the probability calculation: out of 52 cards, 13 are clubs, making these your favorable outcomes.

Identifying favorable outcomes clearly is crucial as it determines the numerator in the probability fraction, serving as a cornerstone of probability calculations.

Probability Formula

The probability formula is a fundamental tool in mathematical statistics used to quantify likelihood. It's generally defined as the ratio of favorable outcomes to the total number of possible outcomes. Here is the basic formula:\[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

• In our card example, the number of favorable outcomes is the 13 clubs in the deck, and the total number of outcomes is all 52 cards.
• When calculated, \( P(\text{Club}) = \frac{13}{52} \), which reduces to \( \frac{1}{4} \).
• This means, when you draw a card from a full deck, there's a 25% probability that it will be a club.

Understanding and applying the probability formula is key to solving a range of problems in probability, allowing us to determine how likely an event is to occur in a given scenario.