Problem 7 - Algebra 2 and Trigonometry

Question

A standard deck of cards contains 52 cards divided into 4 suits. There are two red suits: hearts and diamonds, and two black suits: clubs and spades. Each suit contains 13 cards; ace, king, queen, jack, and cards numbered 2 through \(10 .\) A card is drawn from a standard deck without replacement. What is the probability that the card is a king?

Step-by-Step Solution

Answer

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

Step 1: Determine the Total Number of Cards

A standard deck of cards contains 52 cards. This is the total number of possible outcomes when drawing one card.

Step 2: Identify the Number of Kings

In a standard deck, there are 4 suits: hearts, diamonds, clubs, and spades. Each suit contains one king, so there are 4 kings in total.

Step 3: Calculate the Probability of Drawing a King

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For drawing a king, the probability is \( \frac{4}{52} \).

Step 4: Simplify the Probability Fraction

Simplifying \( \frac{4}{52} \) results in \( \frac{1}{13} \). This is achieved by dividing both the numerator and the denominator by 4.

Key Concepts

Understanding a Standard Deck of Cards

A standard deck of cards is a common tool used in games, magic tricks, and mathematical problems. It consists of 52 total cards.
These cards are equally divided into four groups, known as suits. Each suit contains exactly 13 cards, making each equally represented. This uniform distribution plays a key role in calculating probabilities.
A standard deck does not include jokers, which makes counting and probability calculations straightforward.

Exploring Suits: Hearts, Diamonds, Clubs, and Spades

The four suits in a deck of cards are hearts, diamonds, clubs, and spades.
They are split into two colors:

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

Each suit encompasses an ace, number cards from 2 to 10, and three face cards being the jack, queen, and king. Understanding the composition of these suits is essential for calculating probabilities, such as determining the likelihood of drawing a card of a particular rank or suit.

Simplifying a Probability Fraction

One important skill in probability is simplifying fractions. This is the process of reducing a fraction to its lowest terms.
For probability calculations, such as drawing a king, simplifying \( \frac{4}{52} \) to \( \frac{1}{13} \) is a straightforward operation but necessary to express probabilities in the simplest form.
This can often be achieved by finding the greatest common divisor (GCD) of the numerator and denominator. In the case of \( \frac{4}{52} \,\) dividing both by 4 yields \( \frac{1}{13} \,\) simplifying the probability significantly and aiding in clear communication.

Determining Event Outcomes in Probability

In probability, an event outcome refers to the result of a specific action—in this case, drawing a card from a deck.
The probability of any event is calculated as the number of favorable outcomes over the total number of possible outcomes.
For example, when asked about the probability of drawing a king, we consider:

• Total outcomes: the entire deck, or 52 cards.
• Favorable outcomes: the 4 kings present in the deck.

This results in the probability fraction \( \frac{4}{52} \,\) which simplifies to \( \frac{1}{13} \). Understanding these basic principles helps you tackle more complex problems by breaking them into smaller, comprehensible parts.