Problem 15 - College Algebra with Modeling and Visualization

Question

Find the probability of each event. Randomly drawing a king from a standard deck of 52 cards

Step-by-Step Solution

Answer

The probability of drawing a king is \( \frac{1}{13} \).

Step 1: Understanding Probability

Probability is the measure of the likelihood that an event will occur. It is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Step 2: Identifying the Total Number of Outcomes

In a standard deck of 52 cards, every single card is a possible outcome when drawing one card. Thus, the total number of possible outcomes is 52.

Step 3: Identifying Favorable Outcomes

A standard deck has 4 kings (one from each suit: hearts, diamonds, clubs, and spades). These 4 kings are the favorable outcomes.

Step 4: Calculating Probability

The probability of drawing a king is calculated using the formula: \[ P( ext{Event}) = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Number of Possible Outcomes}} \]. In this case, it is \( \frac{4}{52} \).

Step 5: Simplifying the Probability Fraction

Simplify the fraction \( \frac{4}{52} \) by dividing both the numerator and the denominator by the greatest common divisor (4): \[ \frac{4}{52} = \frac{1}{13} \].

Key Concepts

Standard Deck of Cards

Understanding probability with a deck of cards is a fundamental skill in learning about probability. A standard deck consists of 52 playing cards, which are split into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards ranked from Ace to King.

Here’s a quick breakdown:

• Hearts and Diamonds: These suits contain red cards.
• Clubs and Spades: These suits are composed of black cards.
• Each suit includes one Ace, numbers two through ten, and the face cards: Jack, Queen, and King.

When working with card-based probability problems, knowing this basic breakdown helps in accurately counting and categorizing the cards for various outcomes. Since each card type (like a king or a queen) similarly appears once in each suit, standard decks are perfect tools for practicing uniform probability scenarios.

Favorable Outcomes

Favorable outcomes are the specific events we're examining within the context of all possible outcomes. For instance, if we are trying to determine the probability of drawing a king, then the favorable outcomes are any of the king cards out of the total deck.

Since a standard deck of cards has four kings—one from each suit—those are our favorable outcomes. Remember:

• There is 1 king in each suit: the King of Hearts, King of Diamonds, King of Clubs, and King of Spades, making a total of 4 kings.
• These are the key cards we focus on for this problem.

When counting favorable outcomes, it’s vital to know exactly which and how many elements fulfill the condition we are testing. This sets the stage for calculating probability.

Probability Fraction

Calculating the probability of an event involves creating a probability fraction, also referred to as the probability ratio. This fraction shows the likelihood of the event happening based on the number of favorable outcomes over the total number of possible outcomes.

The formula for calculating probability is:\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]In our example, with 4 kings in a deck of 52 cards, the fraction is initially \( \frac{4}{52} \).

To simplify, we reduce this fraction using the greatest common divisor, which is 4 in this case. Divide both the numerator and the denominator by 4, resulting in \( \frac{1}{13} \). This simplified fraction represents the probability of drawing a king from a standard deck.

Simplifying probability fractions makes it easier to understand and compare probabilities visually and numerically, ensuring clarity in probability problems.