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Magazine Chapter 12: Problem 5
A die is rolled. Find each probability. \(P(\text { multiple of } 3 \text { or } 4)\)
Short Answer
Expert verified
The probability is \( \frac{1}{2} \).
Step by step solution
01
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.
02
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.
03
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.
04
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.
05
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.
06
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, \[ P(\text{multiple of 3 or 4}) = \frac{3}{6} = \frac{1}{2} \]
07
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 \( \frac{1}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
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.
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}.
