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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)
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 \(\frac{1}{6}\).
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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