Problem 17 - Thinking Mathematically

Question

In Exercises 17-22, you randomly select one card from a 52-card deck. Find the probability of selecting a 2 or a 3 .

Step-by-Step Solution

Answer

The probability of selecting a 2 or a 3 from a 52-card deck is \(P = \frac{2}{13}\).

Step 1: Determine the Total Possibilities

There are 52 cards total in a standard deck. This is our total possible outcomes, represented as \(N\). So, \(N = 52\).

Step 2: Identify the Number of Desired Outcomes

There are 4 two's and 4 three's in a deck making a total of 8 desired outcomes. Let this be represented as \(n\), so \(n = 8\).

Step 3: Calculate the Probability

The formula for probability is \(P = \frac{n}{N}\). Substitute \(n = 8\) and \(N = 52\) into the formula to get the probability \(P = \frac{8}{52}\). Simplify the fraction to reduce it to lowest terms.

Key Concepts

Combinatorics

Combinatorics helps us count the number of possible arrangements or combinations in a set, which is essential when calculating probabilities. In a typical deck of cards, we often use combinatorics to determine how many ways we can choose different cards or sets of cards. For example, when determining the probability of drawing a 2 or a 3 from a deck, we first use combinatorics to count the desired outcomes. With 4 twos and 4 threes in the deck, there are 8 cards of interest to our situation.

It's important to note the distinction between permutations and combinations in combinatorics. Permutations consider order, while combinations do not. In this card scenario, since we’re simply interested in whether you draw a card that is either a 2 or a 3, the order does not matter, and we focus on combinations.

Combinatorics simplifies complex counting problems, providing a systematic approach to solve problems like card probabilities in a consistent manner.

Card Games

Card games are a practical application area of probability and combinatorics. A standard deck consists of 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 ranks ranging from Ace to King. Understanding this basic structure is crucial when solving probability problems related to card games.

For example, knowing that each suit contains exactly one 2 and one 3 helps determine how our desired outcome is calculated. By recognizing that we have four 2s and four 3s, we easily total the desired outcomes to 8. In card games like poker or bridge, this level of understanding is key for players planning strategy and anticipating probability outcomes based on the visible cards.

Thus, a deep knowledge of the deck's composition and the basic principles of probability can significantly enhance both the enjoyment and performance in card games.

Basic Arithmetic

Basic arithmetic serves as the foundation for calculating probabilities, simplifying ratios, and ensuring clear, error-free calculations. When working with a probability exercise, like finding the chance of drawing a 2 or a 3, basic arithmetic is applied in several steps.

First, counting how many cards fit our criteria (4 twos plus 4 threes gives us 8) uses simple addition. The total number of possible outcomes is 52, which is the total count of cards. We apply arithmetic to find the probability by dividing the number of desired outcomes (8) by the total outcomes (52), resulting in a fraction: \( P = \frac{8}{52} \).

Performing arithmetic can also involve simplifying fractions to make them easier to understand. In this case, dividing both the numerator and the denominator by their greatest common factor of 4 simplifies \( \frac{8}{52} \) to \( \frac{2}{13} \). This is the simplified probability that a randomly selected card is either a 2 or a 3.