You roll two six-sided dice. Find the probability of the event D that the absolute value of the difference of the dice is 3.
You roll two six-sided dice. Find the probability of the event D that the absolute value of the difference of the dice is 3.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Problem Statement:**
You roll two six-sided dice. Find the probability of the event \( D \) that the absolute value of the difference of the dice is 3.
**Solution Explanation:**
1. **Total Possible Outcomes:**
The total number of possible outcomes when rolling two six-sided dice is \( 6 \times 6 = 36 \).
2. **Defining the Event \( D \):**
The event \( D \) occurs when the absolute value of the difference between the numbers on the two dice is 3. This can be represented as:
\[
|x - y| = 3
\]
where \( x \) and \( y \) are the numbers shown on the first and second die, respectively.
3. **Determining Favorable Outcomes:**
To find the favorable outcomes, we list the pairs \((x, y)\) such that the difference is either 3 or -3:
- \( x = 1, y = 4 \)
- \( x = 2, y = 5 \)
- \( x = 3, y = 6 \)
- \( x = 4, y = 1 \)
- \( x = 5, y = 2 \)
- \( x = 6, y = 3 \)
There are 6 favorable outcomes in total.
4. **Calculating the Probability:**
The probability of the event \( D \) is the ratio of the number of favorable outcomes to the total number of possible outcomes:
\[
P(D) = \frac{6}{36} = \frac{1}{6}
\]
**Conclusion:**
The probability that the absolute value of the difference between the numbers on the two dice is 3 is \(\frac{1}{6}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F6dec2e08-a94f-4d85-a43c-33f85cc75cce%2F3u25brk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
You roll two six-sided dice. Find the probability of the event \( D \) that the absolute value of the difference of the dice is 3.
**Solution Explanation:**
1. **Total Possible Outcomes:**
The total number of possible outcomes when rolling two six-sided dice is \( 6 \times 6 = 36 \).
2. **Defining the Event \( D \):**
The event \( D \) occurs when the absolute value of the difference between the numbers on the two dice is 3. This can be represented as:
\[
|x - y| = 3
\]
where \( x \) and \( y \) are the numbers shown on the first and second die, respectively.
3. **Determining Favorable Outcomes:**
To find the favorable outcomes, we list the pairs \((x, y)\) such that the difference is either 3 or -3:
- \( x = 1, y = 4 \)
- \( x = 2, y = 5 \)
- \( x = 3, y = 6 \)
- \( x = 4, y = 1 \)
- \( x = 5, y = 2 \)
- \( x = 6, y = 3 \)
There are 6 favorable outcomes in total.
4. **Calculating the Probability:**
The probability of the event \( D \) is the ratio of the number of favorable outcomes to the total number of possible outcomes:
\[
P(D) = \frac{6}{36} = \frac{1}{6}
\]
**Conclusion:**
The probability that the absolute value of the difference between the numbers on the two dice is 3 is \(\frac{1}{6}\).
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