Problem 12 - 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. \(P(\) King \(\cap\) Red \()\)

Step-by-Step Solution

Answer

Answer: The probability of drawing a red king card from a standard deck is 1 out of 26.

Step 1: Identify the red king cards in the deck

In a standard deck, red cards consist of hearts and diamonds, and there is one king in each suit. So, there are 2 red king cards: the King of Hearts and the King of Diamonds.

Step 2: Determine the total number of cards in the deck

A standard deck consists of 52 cards: 13 cards in each of the four suits (hearts, diamonds, clubs, and spades).

Step 3: Calculate the probability

We are looking for the probability of drawing a red king card from the deck. We know that there are 2 red king cards and a total of 52 cards in the deck. Thus, the probability of drawing a red king card is given by the number of red king cards divided by the total number of cards in the deck. The probability is: \(\frac{\text{number of red king cards}}{\text{total number of cards}} = \frac{2}{52}\)

Step 4: Simplify the fraction

To simplify the fraction, we note that both the numerator and the denominator are divisible by 2: \(\frac{2}{52} = \frac{1}{26}\)

Step 5: State the probability in words

The probability of drawing a card from a standard deck that is both a king and red is 1 out of 26.

Key Concepts

Probability of Compound Events

When discussing the probability of compound events, we refer to the likelihood of two or more events happening together. In the context of card games, this often involves finding the probability of drawing specific combinations of cards. For example, drawing a card that is both a King and red can be viewed as a compound event.

• A compound event means the occurrence of event A and event B together. Here, event A is drawing a King and event B is drawing a red card.
• The probability of compound events is often calculated by identifying the overlap of these events. In our example, it would involve recognizing which Kings are also red.

To find this probability, we calculate how frequently this combination occurs and divide it by the total number of possible outcomes. Compound events are central to solving problems where multiple criteria need to be met at once.

Sample Space in Probability

The sample space in probability encompasses all possible outcomes of a particular experiment or situation. In card games, the sample space is the set of all cards in the deck. For a standard deck, this involves:

• 52 cards total
• 4 suits: hearts, diamonds, clubs, and spades
• Each suit contains 13 cards

Understanding the sample space is crucial, as it helps to determine the denominator of the probability fraction. In our problem, the total number of cards (52) in the deck serves as our sample space. This foundational knowledge helps when dealing with probability questions, ensuring that we are considering all potential results. By knowing the entire collection of possibilities, calculations of probabilities can be more accurate and meaningful.

Simplifying Fractions in Probability

Simplifying fractions in probability ensures that the answer is presented in its most basic form, making it easier to interpret. When you calculate probability, such as the chance of drawing a red king, you initially find an unsimplified fraction: \[\frac{2}{52}\]This fraction arises because there are 2 favorable outcomes (red kings) out of 52 possible outcomes (all cards). To simplify, you check for any common factors between the numerator and the denominator. In this example:

• Both numbers are divisible by 2.
• So we divide both by 2 to get \(\frac{1}{26}\).

By simplifying, we better understand the probability: 1 out of 26. This also helps in comparing probabilities between different events, as a reduced fraction provides a clearer picture of the likelihood of an occurrence.