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. \(P\) (Heart \(\mid\) Red)

Short answer

Answer: The meaning of the probability expression \(P\)(Heart \(\mid\) Red) is "The probability of drawing a Heart card given that the card drawn is Red." The probability as a fraction is \(\frac{1}{2}\).

Step-by-step solution

Understand the given probability expression and state its meaning in words.

The given probability expression is \(P\)(Heart \(\mid\) Red). This expression represents the conditional probability of drawing a Heart card given that the card drawn is Red. In other words, if we know that the card we have drawn is Red, what is the probability that it is a Heart card?

Identify the total number of Red cards and Heart cards in a standard deck.

In a standard deck of 52 cards, there are 26 Red cards (13 cards each of Hearts and Diamonds suits) and 13 Heart cards (all of them are Red). So, there are 13 Red Heart cards and 26 Red cards in total.

Calculate the probability as a fraction.

We have 13 Red Heart cards and 26 total Red cards. We can now calculate the probability of drawing a Heart card given the card is Red as follows: $$ P(\text{Heart} \mid \text{Red}) = \frac{\text{Number of Red Heart cards}}{\text{Number of Red cards}} = \frac{13}{26} $$ Now, simplify the fraction: $$ P(\text{Heart} \mid \text{Red}) = \frac{13}{26} = \frac{1}{2} $$

Present the final answer.

The meaning of the probability expression \(P\)(Heart \(\mid\) Red) is "The probability of drawing a Heart card given that the card drawn is Red." The probability as a fraction is \(\frac{1}{2}\).

Key concepts

Understanding Probability

Probability is a measure that describes how likely an event is to occur.
It is expressed as a number between 0 and 1, where 0 means the event cannot happen, and 1 indicates certainty.
Conditional probability refers to the probability of an event occurring, given that another event has already happened.
In our exercise, we deal with conditional probability: \(P(\text{Heart} \mid \text{Red})\).
This reads as 'the probability of drawing a Heart card given that a Red card is drawn'.
It's crucial to understand that we're narrowing our sample space to only Red cards.
With conditional probability, the previous known outcomes affect our likelihood calculation.

• Probability helps you predict outcomes based on current knowledge.
• Conditional probability assesses likelihood with an updated scenario.
• The notation \(P(A \mid B)\) means 'probability of A given B'.

In this exercise, knowing you have a Red card impacts the probability of it being a Heart card.

The Art of Fraction Simplification

Simplifying fractions makes calculations easier and results more understandable.
In mathematics, simplification involves reducing a fraction to its smallest form.
For instance, \( \frac{13}{26} \) simplifies to \( \frac{1}{2} \).
This transformation is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).
This makes comparing and working with fractions more manageable in problems.

• Find the GCD of the numerator and denominator.
• Divide both by the GCD to simplify the fraction.
• Simplify whenever possible for clarity and ease of understanding.

When fractions are simplified, they're cleaner and easier to interpret, avoiding any unnecessary complexity in your calculations.

The Standard Deck of Cards Explained

A standard deck of cards is a familiar tool in probability and statistics exercises.
It consists of 52 cards, divided into four suits: Hearts, Diamonds, Clubs, and Spades.
Each suit has 13 cards ranging from Ace through 10, and includes three face cards: Jack, Queen, and King.

• Two suits are Red: Hearts and Diamonds.
• Two suits are Black: Clubs and Spades.
• Understanding the composition of a deck helps in probability problems.

Knowing this structure allows you to predict outcomes based on different constraints.
For example, saying 13 out of the 26 Red cards are Hearts helps deduce the probability of Heart given a Red card.
Each standard deck is uniform, meaning each card draw is independent and equally likely within its constraints.