Problem 28 - College Algebra

Question

For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: Six or seven

Step-by-Step Solution

Answer

The probability of drawing a six or seven is \( \frac{2}{13} \).

Step 1: Understand the Total Number of Possible Outcomes

A standard deck of cards has a total of 52 cards. When a single card is drawn, there are 52 possible outcomes. Therefore, the total number of possible outcomes is 52.

Step 2: Identify the Desired Outcomes

We need to find the probability of drawing a six or a seven. In a deck, there are 4 sixes (one for each suit) and 4 sevens (one for each suit). Therefore, the total number of favorable outcomes (drawing a six or a seven) is 4 sixes + 4 sevens = 8.

Step 3: Calculate the Probability

The probability of an event occurring is given by the formula: \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \). For drawing a six or a seven, this is \( \frac{8}{52} \).

Step 4: Simplify the Fraction

Simplify \( \frac{8}{52} \) by dividing both numerator and denominator by their greatest common divisor, which is 4. This becomes \( \frac{2}{13} \). Therefore, the probability of drawing either a six or a seven from a standard deck of cards is \( \frac{2}{13} \).

Key Concepts

Standard Deck of Cards

To understand card-related probability, it's crucial to familiarize yourself with a standard deck of cards. A full set consists of 52 unique cards. Each card is categorized into four suits: hearts, diamonds, clubs, and spades.

• **Hearts** and **diamonds** are typically red, while **clubs** and **spades** are black.
• Each suit contains 13 cards, ranging from numbers 2 through 10, and also including three face cards: Jack, Queen, and King, plus an Ace.

This setup means each number or face value is represented once per suit. For probabilities that concern particular card values or sets, like sixes or sevens, keep in mind that each specific value appears four times. Understanding this basic structure of a deck helps simplify and accurately determine the probability calculations.

Favorable Outcomes

When calculating probability, identifying favorable outcomes is a key step. Favorable outcomes are the specific results that match the event you are interested in.

• For example, if you're looking for the probability of drawing a six or a seven from a standard deck, these are your favorable outcomes.
• In this case, there are 4 sixes and 4 sevens—one of each in hearts, diamonds, clubs, and spades. Thus making a total of 8 favorable outcomes.

By clearly identifying the favorable outcomes, we set the stage to calculate probabilities by comparing these outcomes to the total number of outcomes possible. Always ensure that you don’t double-count or miss any possibilities when listing favorable outcomes.

Simplifying Fractions

Once you have found the probability as a fraction, it’s good practice to simplify it. Simplifying makes it easier to interpret and communicate the probability.

• The probability is defined as the ratio of favorable outcomes to the total number of possible outcomes.
• For drawing a six or a seven, the initial fraction is \( \frac{8}{52} \).

To simplify \( \frac{8}{52} \), you find the greatest common divisor (GCD) of both the numerator (8) and the denominator (52). The GCD here is 4.
By dividing both 8 and 52 by 4, the fraction simplifies to \( \frac{2}{13} \). This is the simplest form, making it clear and concise to express the chance of drawing either a six or a seven.