Problem 5 - Algebra 2

Question

A die is rolled. Find each probability. \(P(\text { multiple of } 3 \text { or } 4)\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{2} \).

Step 1: Understand the Problem

We are asked to find the probability of rolling a number on a die that is either a multiple of 3 or a multiple of 4. A standard die has six faces numbered 1 through 6.

Step 2: Identify Multiples of 3

Examine the numbers on the die to find which are multiples of 3. The multiples of 3 within the numbers 1 to 6 are 3 and 6. Thus, there are two numbers that are multiples of 3.

Step 3: Identify Multiples of 4

Next, determine the numbers that are multiples of 4. Looking at the numbers from 1 to 6, only the number 4 is a multiple of 4.

Step 4: Identify Overlapping Numbers

Now, check if there are any numbers that are both multiples of 3 and multiples of 4. In this case, there are no such overlapping numbers in the range from 1 to 6.

Step 5: Calculate the Total Number of Favorable Outcomes

The numbers that satisfy our condition (multiple of 3 or 4) are 3, 4, and 6. Hence, there are three favorable outcomes overall.

Step 6: Calculate the Probability

The probability of rolling a multiple of 3 or 4 is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. The formula for probability is:\[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]So, \[ P(\text{multiple of 3 or 4}) = \frac{3}{6} = \frac{1}{2} \]

Step 7: Simplify the Probability

To simplify the probability, divide both the numerator and denominator by their greatest common divisor, which is 3 in this case. This gives us the simplified probability of \( \frac{1}{2} \).

Key Concepts

Understanding Multiples

A multiple is a number that can be divided by another number without leaving a remainder. In the case of rolling a die, we examine each of the six faces to determine which numbers are multiples of a given number, such as 3 or 4.

To find multiples of 3, check the numbers on the die: 1 through 6. Only 3 and 6 meet the criteria. Similarly, the multiples of 4 from the numbers available are found by dividing each by 4. In this scenario, only the number 4 itself is a multiple of 4.

• Multiples are determined by division with no remainder.
• A face on a die is a multiple of a number if it can be divided exactly by that number.
• 3 and 6 are multiples of 3, while 4 is a multiple of 4.

Recognizing multiples helps us narrow down the possible numbers that meet our conditions.

Understanding Outcomes

Outcomes in probability refer to the possible results that could occur from a random event. When rolling a six-sided die, the outcomes are the numbers that appear face-up: 1, 2, 3, 4, 5, and 6.

In our exercise, we focus specifically on the outcomes that are either multiples of 3 or multiples of 4. We identified these outcomes as 3, 4, and 6.

• Each face of the die represents an outcome when rolled.
• Outcomes can be filtered based on specific criteria (like being a multiple).
• In our scenario, the favorable outcomes are 3, 4, and 6 because they meet the conditions.

Counting these favorable outcomes is crucial to determining the probability we are interested in.

Simplification in Probability

Simplification is a crucial step in handling probability because it makes numbers easier to interpret and use. For probability, simplification often involves reducing a fraction to its simplest form.

When calculating the probability of interest, we find that the initial probability rac{3}{6} can be simplified. We do this by finding the greatest common divisor (GCD) of the numerator (3) and denominator (6), which is 3.

• Divide both the numerator and denominator by their GCD to simplify the fraction.
• The original probability rac{3}{6} simplifies to rac{1}{2}.

Simplifying probability fractions helps provide a clearer understanding of the likelihood of an event occurring, making it easier to grasp probabilities at a glance.