Problem 11 - 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. $$ P(\operatorname{Red} \cap \text { King }) $$

Step-by-Step Solution

Answer

Answer: The probability expression \(P(\operatorname{Red} \cap \text { King })\) represents the probability of drawing a red king (both red and a king) from a standard deck of cards. The probability as a fraction is \(\frac{1}{26}\).

Step 1: Understand the meaning of the expression in words

The expression \(P(\operatorname{Red} \cap \text { King })\) represents the probability of drawing a red king from a standard deck of cards. In other words, it's the probability of picking a card that is both red (hearts or diamonds) and a king.

Step 2: Calculate the number of red kings and total cards in the deck

In a standard deck of 52 cards, there are 4 suits: hearts, diamonds, spades, and clubs. Each suit has 13 cards: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. Since hearts and diamonds are red suits, there are 2 red kings (King of Hearts and King of Diamonds) in the deck. So, there are 2 red kings out of the 52 cards.

Step 3: Calculate the probability as a fraction

To determine the probability as a fraction, divide the number of red kings by the total number of cards in the deck: $$ P(\operatorname{Red} \cap \text { King }) = \frac{\text{Number of red kings}}{\text{Total number of cards}} = \frac{2}{52} $$ Simplify the fraction: $$ P(\operatorname{Red} \cap \text { King }) = \frac{1}{26} $$ So, the probability of drawing a red king from a standard deck of cards is \(\frac{1}{26}\).

Key Concepts

Standard Deck of Cards

A standard deck of cards is a deck consisting of 52 cards, often used in a variety of card games. This deck is evenly divided into four suits: hearts, diamonds, clubs, and spades. Each suit consists of 13 cards, ranging from Ace to King. In a standard deck:

• Each suit includes one Ace, numbered cards from 2 to 10, and three face cards (Jack, Queen, King).
• The suits are equally distributed, so each suit has an identical number of cards.

Such a distribution ensures that any card type or suit has an equal probability of being drawn, given they are in equal numbers.

Red Suits

In a standard deck of cards, the red suits refer to the hearts and diamonds. These suits are visually distinct with a red color for easy identification. Key characteristics of red suits include:

• Each red suit contains 13 cards, making a total of 26 red cards out of the full 52-card deck.
• Among the red cards, there are one Jack, one Queen, and one King per suit.

Understanding the composition of red suits is important when calculating probabilities involving red cards, like when determining the likelihood of drawing a red king.

Probability Fraction

Probability is a way to quantify the likelihood of an event happening, usually expressed as a fraction, decimal, or percentage. In card games, probability helps predict outcomes such as drawing a specific card. To find the probability as a fraction, follow these steps:

• Identify the number of successful outcomes (for instance, the number of red kings).
• Divide this number by the total number of possible outcomes (total number of cards).

For example, when calculating the probability of drawing a red king, you find there are 2 red kings in a 52-card deck:\[\text{Probability} = \frac{\text{Number of red kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}\]This fraction represents the chance of picking a red king from a well-shuffled deck.