Problem 25 - Algebra for College Students

Question

Find the probability of each of the following events:A spade or a diamond is drawn. \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{2} \).

Step 1: Understanding the Question

We need to find the probability that a drawn card from a standard deck of 52 cards is either a spade or a diamond.

Step 2: Calculate Total Possible Outcomes

A standard deck has 52 cards, and each card drawing is an equally likely event. Thus, the total number of possible outcomes is 52.

Step 3: Identify Favorable Outcomes

There are 13 spades and 13 diamonds in a deck of cards, so there are 13 + 13 = 26 favorable outcomes for drawing either a spade or a diamond.

Step 4: Apply Probability Formula

The probability of an event is given by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability of drawing a spade or a diamond is \( \frac{26}{52} \).

Step 5: Simplify the Fraction

The fraction \( \frac{26}{52} \) can be simplified by dividing both the numerator and the denominator by 26, resulting in \( \frac{1}{2} \).

Key Concepts

Probability Calculation

Probability is a fundamental concept in mathematics. It measures how likely an event is to occur. The probability of an event can be calculated using a simple formula: the number of favorable outcomes divided by the total number of possible outcomes.

To further understand this, let's break it down:

• Favorable Outcomes: These are the outcomes that satisfy the condition of the event we're interested in.
• Total Possible Outcomes: This is the set of all possible outcomes in the scenario.

In the exercise, we calculate the probability of drawing a spade or a diamond from a deck of cards. Hence, the number of favorable outcomes (drawing a spade or a diamond) is 26, as there are 13 spades and 13 diamonds. The total number of outcomes is the 52 cards in the deck. Thus, the probability calculation is done by dividing 26 by 52, simplifying to \(\frac{1}{2}\). This tells us that there's an equal chance of drawing a spade or a diamond when pulling a single card from a shuffled deck.

Standard Deck of Cards

A standard deck of cards is a familiar tool used in many probability exercises. Knowing what's inside this deck helps in solving probability problems effectively.

Here's a breakdown of the composition of a standard deck:

• It contains a total of 52 cards.
• These cards are divided into 4 suits: spades, hearts, diamonds, and clubs.
• Each suit consists of 13 cards: Ace through 10, and three face cards—Jack, Queen, and King.

Understanding this structure helps determine both total possible outcomes and favorable outcomes in probability exercises efficiently. In our original problem, knowing the deck's makeup reveals why drawing a spade or a diamond results in an event probability of \(\frac{1}{2}\). It guides in seeing how equally distributed the suits are within the deck.

Favorable Outcomes

When calculating probability, identifying favorable outcomes is a key step. Favorable outcomes are those specific results that are desired or are being calculated.

In the context of a card deck, if we are asked to find the probability of drawing a spade or a diamond, identifying favorable outcomes means counting each spade and each diamond in the deck.

• Spades and Diamonds each have 13 cards in a standard deck.
• So, the total number of favorable outcomes when drawing either a spade or a diamond is 13 + 13 = 26.

The accuracy in determining these outcomes directly influences the correctness of our probability calculations. By ensuring all favorable outcomes are counted, we establish the numerator of our probability fraction, ensuring an accurate probability result.