Problem 24 - Algebra for College Students

Question

Find the probability of each of the following events:A king is drawn. \(\frac{1}{13}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Setup

A standard deck of playing cards has 52 cards comprising four different suits: hearts, diamonds, clubs, and spades. Each suit contains 13 ranks: Ace, 2, 3, ..., 10, Jack, Queen, and King.

Step 2: Identify the Successful Outcomes

To find the probability of drawing a king, we must identify how many kings are available in a deck. Each suit has exactly one king, so there are a total of 4 kings in a deck.

Step 3: Calculate Total Possible Outcomes

The total number of possible outcomes when drawing a single card from a deck is equal to the total number of cards in the deck, which is 52.

Step 4: Calculate Probability

Probability is calculated by dividing the number of successful outcomes by the total number of possible outcomes. The probability of drawing a king is calculated as follows:\[P(\text{King}) = \frac{\text{Number of Kings}}{\text{Total Number of Cards}} = \frac{4}{52}\]

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{4}{52}\) by dividing the numerator and the denominator by their greatest common divisor, which is 4:\[\frac{4}{52} = \frac{4 \div 4}{52 \div 4} = \frac{1}{13}\]

Key Concepts

Deck of Cards

A standard deck of cards is a fascinating concept used in probability calculations and games worldwide. This deck consists of 52 cards divided equally among four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 ranks, which are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. Understanding this composition is crucial for probability exercises, as each card represents a different potential outcome. The symmetry and balance in the deck make it an ideal tool for teaching probability concepts. When dealing with these cards, remember the following key points:

• Ace through 10 are the numbered cards.
• Jack, Queen, and King are commonly referred to as the face cards.
• Each suit—hearts, diamonds, clubs, spades—includes one King, for a total of four Kings in the entire deck.

These details help in identifying potential outcomes and successful events when attempting to calculate probabilities.

Successful Outcomes

In probability, identifying successful outcomes is a critical step in solving a problem. A 'successful outcome' is the scenario we are interested in, for which we are trying to find the probability. When drawing a card from a deck and looking for a King, the successful outcomes are specifically the four Kings present in the deck. This process involves counting how many instances of the desired result are available. For example:

• There is one King in each suit.
• Therefore, there are four Kings in total—one in hearts, one in diamonds, one in clubs, and one in spades.

The concept of successful outcomes is about pinpointing these specific scenarios within the total set of possibilities. This focus allows us to calculate the likelihood of an event, such as drawing a King, by comparing it to the total number of possibilities.

Probability Calculation

Calculating probability involves a simple yet powerful formula that helps determine the likelihood of a specific event. The formula used is the ratio of the number of successful outcomes to the total number of possible outcomes. When calculating the probability of drawing a King from a standard deck of 52 cards, here's how you go about it:

• First, count the total number of successful outcomes (4 Kings).
• Next, identify the total number of possible outcomes, which is the total number of cards in the deck (52).

Using these numbers, the probability of drawing a King is calculated as follows:\[ P(\text{King}) = \frac{\text{Number of Kings}}{\text{Total Number of Cards}} = \frac{4}{52} \]Finally, simplify the fraction to its simplest form. In this case, \(\frac{4}{52}\) simplifies to \(\frac{1}{13}\), since both the numerator and denominator can be divided by 4. Understanding this process is essential in mastering basic probability and can be applied to a wide range of scenarios beyond playing cards.