Problem 14 - 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 \(\mid\) King \()\)

Step-by-Step Solution

Answer

Answer: The probability is 1/2 or 50%.

Step 1: Understand the problem

We are given a standard deck of 52 playing cards. There are 26 red cards (13 hearts and 13 diamonds) and 26 black cards (13 clubs and 13 spades). We also have 4 Kings (1 in each suit). We want to find the probability that if we draw a King, that King is a red card.

Step 2: Formula for conditional probability

We will use the formula for conditional probability, which is given by \(P(A\mid B) = \frac{P(A \cap B)}{P(B)}\), where \(A\) is the event that we draw a red card and \(B\) is the event that we draw a King. We need to find \(P(\text{Red} \mid \text{King})\), so we need to calculate \(P(\text{Red} \cap \text{King})\) and \(P(\text{King})\).

Step 3: Calculate P(Red ∩ King) and P(King)

Let's first calculate \(P(\text{Red} \cap \text{King})\). There are 2 Kings that are red: one from hearts and one from diamonds. So, there are 2 cards that are both red and Kings. Therefore, \(P(\text{Red} \cap \text{King}) = \frac{2}{52}\). Now, let's calculate \(P(\text{King})\). There are 4 Kings in total: one from each suit. Therefore, \(P(\text{King}) = \frac{4}{52}\).

Step 4: Calculate P(Red ∣ King)

Now we can use the formula for conditional probability: \(P(\text{Red} \mid \text{King}) = \frac{P(\text{Red} \cap \text{King})}{P(\text{King})} = \frac{\frac{2}{52}}{\frac{4}{52}} = \frac{2}{4} = \frac{1}{2}\).

Step 5: Interpret the result

The expression \(P(\text{Red} \mid \text{King})\) represents the probability of drawing a red card, given that the card we draw is a King. The probability is \(\frac{1}{2}\), which means that if we draw a King, there is a 50% chance that it will be a red card.

Key Concepts

Standard Deck of Cards

A standard deck of cards is a collection of 52 playing cards. These cards are divided into four suits:

• Hearts
• Diamonds
• Clubs
• Spades

Each suit contains 13 cards, including the numbers 2 through 10, as well as a Jack, Queen, King, and Ace. The suits can be further categorized by color:

• Red Suits: Hearts and Diamonds
• Black Suits: Clubs and Spades

This means that there are 26 red cards (13 hearts and 13 diamonds) and 26 black cards (13 clubs and 13 spades) in a standard deck. Understanding the composition of a standard deck is crucial when calculating probabilities involving cards since each card in the deck has an equal chance of being drawn.

Probability Fraction

Probability is a measure of how likely an event is to occur. When dealing with probabilities, we often express them as fractions. A probability fraction is computed by comparing the number of favorable outcomes to the total number of possible outcomes. For example, if you want to compute the probability fraction for drawing a King from a standard deck, you look at the number of Kings in the deck (which is 4) and divide it by the total number of cards (which is 52). This gives us a probability fraction:\[P(\text{King}) = \frac{4}{52} = \frac{1}{13}\]. In the context of this exercise, probability fractions allow us to succinctly represent the likelihood of events involving playing cards, such as finding the chance of drawing a red card, given that it’s a King.

Event Intersection

An event intersection in probability refers to scenarios where two events both occur. In mathematical terms, it's denoted as \(A \cap B\), meaning that both events A and B happen. In the context of the card problem, consider the intersection of the events "drawing a red card" and "drawing a King." If we denote "Red card" as event A and "King" as event B, the intersection \(A \cap B\) includes only the red Kings. Understanding event intersections is essential when solving conditional probability problems. For the card exercise, knowing how many cards satisfy both conditions (being a red card and a King) helps in calculating the conditional probability using the formula: \[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]. Here, it shows the likelihood of drawing a red King when you know the card is already a King.