Problem 10 - 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. $$ P(\text { Red }) $$

Step-by-Step Solution

Answer

Answer: The expression \(P(\text { Red })\) represents the probability of drawing a red card from a standard deck of cards. The probability of drawing a red card is \(\frac{13}{26}\).

Step 1: Understanding the Expression

The given expression, \(P(\text { Red })\), represents the probability of drawing a red card from a standard deck of cards. In a standard deck, there are two colors: red and black. The red cards consist of hearts and diamonds, while black cards consist of clubs and spades.

Step 2: Total Number of Cards in a Standard Deck

A standard deck of cards has 52 cards in total, with 13 cards for each of the four suits (hearts, diamonds, clubs, and spades).

Step 3: Number of Red Cards

Since there are two red suits (hearts and diamonds), and each suit has 13 cards, there are a total of 2 * 13 = 26 red cards in a standard deck.

Step 4: Calculating the Probability

To calculate the probability of drawing a red card, we need to find the fraction of red cards in the deck. Since there are 26 red cards and 52 cards total, the probability of drawing a red card is: $$ P(\text { Red }) = \frac{\text {number of red cards}}{\text {total number of cards}} = \frac{26}{52} $$

Step 5: Simplifying the Probability

To simplify the fraction, we can divide both the numerator and the denominator by the greatest common divisor, which is 2 in this case. So the simplified probability is: $$ P(\text { Red }) = \frac{26\div 2}{52\div 2} = \frac{13}{26} $$

Step 6: Final Answer

The probability expression \(P(\text { Red })\) means the probability of drawing a red card from a standard deck of cards. The probability of drawing a red card is \(\frac{13}{26}\).

Key Concepts

Standard Deck of Cards

A standard deck of cards is a collection of 52 playing cards. These 52 cards are divided equally into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, which include the numbers 2 through 10, and face cards: a jack, queen, king, and an ace. Understanding the structure of a standard deck is crucial for calculating probabilities, especially in card games and probability exercises. Knowing that there are equal numbers of each suit helps simplify probability calculations, as you often deal with either the entire deck or specific sections based on suit or color. For instance, when determining the probability of drawing a red card, it's helpful to know that the hearts and diamonds make up half the deck. This knowledge allows you to easily identify and count the group of interest without having to recount each time.

Simplifying Fractions

Simplifying fractions is a key skill in mathematics, making expressions easier to understand and work with. To simplify a fraction, divide both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). For example, consider the fraction \( \frac{26}{52} \). Both 26 and 52 can be divided by the GCD, which is 26 in this case, simplifying the fraction to \( \frac{13}{26} \). Always ensure the fraction is in its simplest form, meaning no further division of both the numerator and denominator is possible by any common factor other than 1. Simplifying fractions not only makes calculations neater but also helps in understanding the relative size or probability more intuitively. It is especially beneficial in probability exercises, providing a clearer sense of how likely an event is to occur.

Suits of Cards

The suits of cards are the categories into which the cards in a deck are divided. In a standard deck, there are four suits: hearts, diamonds, clubs, and spades. Each suit features its own distinct symbol and color:

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

Being familiar with the suits helps in understanding how a card deck is organized, as well as in probability calculations which often rely on suit-based groupings. For instance, when asked to calculate the probability of drawing a heart or a diamond, knowing these two are red suits guides you in identifying the total number of relevant cards. Each suit contains one quarter of the entire deck, approximately 13 cards, forming not only the basis for many games but also helping in straightforward probability estimates, as you often deal with equal groups.