Problem 16 - Algebra Form and Function

Question

The probability expressions refer to drawing a card from a standard deck of cards. State in words the meaning of the expression and give the probability as a fraction. \(P(\) Red \(\mid\) Heart \()\)

Step-by-Step Solution

Answer

Answer: The probability of drawing a red card given that it is a heart is 1, or 100%.

Step 1: Interpret the Probability Expression

The given expression is \(P(\) Red \(\mid\) Heart \()\). This represents the conditional probability of drawing a red card given that the card is a heart.

Step 2: Identify the Relevant Cards in the Deck

There are 52 cards in a standard deck, with 13 cards in each of the 4 suits (hearts, diamonds, clubs, and spades). The red cards consist of hearts and diamonds, while the black cards consist of clubs and spades. Since we are given the condition that the card is a heart, we are only concerned with the 13 heart cards in the deck.

Step 3: Determine the Number of Red Cards Given the Condition

Given that the card is a heart, all 13 heart cards in the deck are red because hearts and diamonds are the red suits. Therefore, there are 13 red cards given the condition that the card is a heart.

Step 4: Calculate the Probability as a Fraction

To find the probability of drawing a red card given that the card is a heart, we can use the following formula: \(P(\) Red \(\mid\) Heart \() = \frac{\text{Number of red cards given the condition}}{\text{Total number of cards given the condition}}\) So, \(P(\) Red \(\mid\) Heart \() = \frac{13}{13}\)

Step 5: Simplify the Fraction

The fraction \(\frac{13}{13}\) represents the probability that a card is red given that it is a heart. Since 13 goes into itself exactly 1 time, we can simplify the fraction to: \(P(\) Red \(\mid\) Heart \() = 1\) Therefore, the probability of drawing a red card given that it is a heart is 1, or 100%.

Key Concepts

Probability Theory

Probability theory is all about measuring the likelihood of an event happening. In simple terms, it allows us to calculate how probable it is for a specific outcome to occur.
It helps in understanding and quantifying situations involving uncertainty. In this case, we're dealing with conditional probability, which is a type of probability that calculates the chance of an event occurring given that another event has already occurred.
Conditional probability is expressed as \( P(A \mid B) \), which means the probability of event \( A \) occurring given that \( B \) has happened. This concept is crucial in various fields such as statistics, finance, and everyday decision-making.

Deck of Cards

A standard deck of cards is a common tool used in probability exercises.
This deck consists of 52 cards in total. Understanding the structure of the deck is essential to solve probability problems effectively.
Here are some key features of a standard deck:

• Four suits: hearts, diamonds, clubs, and spades.
• Each suit contains 13 cards, ranging from Ace to King.
• Hearts and diamonds are red, while clubs and spades are black.

This structure helps us identify and classify cards based on different criteria, which is fundamental when determining probabilities.

Red and Black Cards

In a standard deck, cards are categorized by color as well as suit. This distinction is often crucial in probability questions.
There are 26 red cards and 26 black cards in a deck.

• Red cards include all the hearts and diamonds.
• Black cards include all the clubs and spades.

Understanding red and black card grouping is necessary, especially in exercises like determining the probability of drawing a particular color card given a certain condition. For example, as seen in the exercise, all heart cards also being red makes it straightforward to calculate related probabilities.

Fraction Simplification

In probability, we often express outcomes as fractions. Simplifying these fractions makes them easier to understand and interpret.
Simplification involves reducing a fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).
In our exercise, we derived a fraction \( \frac{13}{13} \) for the probability, which simplifies to 1. This shows that the event is certain to occur, representing 100% probability.
Here are general steps to simplify fractions:

• Identify the GCD of the numerator and denominator.
• Divide both by the GCD.
• Write the simplified fraction.

Simplification is a powerful tool, making probabilities more intuitive and clearly communicating how likely an event is to happen.