Two fair six-sided dice are rolled. Let A = the event that the dice add to 7, and let C = the event that the dice show the same number. Find P(CIA') and P(C'IA'). P(CIA') = (Simplify your answer.) P(C'A') = (Simplify your answer.)
Two fair six-sided dice are rolled. Let A = the event that the dice add to 7, and let C = the event that the dice show the same number. Find P(CIA') and P(C'IA'). P(CIA') = (Simplify your answer.) P(C'A') = (Simplify your answer.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Probability of Events Involving Two Fair Six-Sided Dice**
**Introduction:**
In this exercise, we will explore the probability of specific events occurring when two fair six-sided dice are rolled. Let us define two events:
- Let \( A \) be the event that the sum of the numbers on the dice equals 7.
- Let \( C \) be the event that the numbers on the dice show the same value.
We will be calculating two conditional probabilities:
1. \( P(C \mid A') \)
2. \( P(C' \mid A') \)
where \( A' \) represents the complement of event \( A \), meaning the event that the sum of the dice does not equal 7, \( C' \) represents the complement of event \( C \), meaning the event that the dice do not show the same number.
**Problem Statement:**
Two fair six-sided dice are rolled. Let \( A \) be the event that the dice add to 7, and let \( C \) be the event that the dice show the same number. Find \( P(C \mid A') \) and \( P(C' \mid A') \).
**Solution:**
1. \( P(C \mid A') = \) [Simplify your answer]
2. \( P(C' \mid A') = \) [Simplify your answer]
**Explanation:**
To solve these problems, we'll need to use the definition of conditional probability:
\[ P(C \mid A') = \frac{P(C \cap A')}{P(A')} \]
\[ P(C' \mid A') = \frac{P(C' \cap A')}{P(A')} \]
Start by calculating the individual probabilities and then use the formulas above to find the conditional probabilities.
---
*Note to educators: This text is designed to provide step-by-step instruction to help students understand how to calculate and simplify the conditional probabilities involving two events when rolling two fair six-sided dice.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff138e77-4b21-4b3f-a581-600ea0faba85%2Ff04b4e77-eb15-493b-89ce-fa0ba3a04de7%2Fv1nhbn_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Probability of Events Involving Two Fair Six-Sided Dice**
**Introduction:**
In this exercise, we will explore the probability of specific events occurring when two fair six-sided dice are rolled. Let us define two events:
- Let \( A \) be the event that the sum of the numbers on the dice equals 7.
- Let \( C \) be the event that the numbers on the dice show the same value.
We will be calculating two conditional probabilities:
1. \( P(C \mid A') \)
2. \( P(C' \mid A') \)
where \( A' \) represents the complement of event \( A \), meaning the event that the sum of the dice does not equal 7, \( C' \) represents the complement of event \( C \), meaning the event that the dice do not show the same number.
**Problem Statement:**
Two fair six-sided dice are rolled. Let \( A \) be the event that the dice add to 7, and let \( C \) be the event that the dice show the same number. Find \( P(C \mid A') \) and \( P(C' \mid A') \).
**Solution:**
1. \( P(C \mid A') = \) [Simplify your answer]
2. \( P(C' \mid A') = \) [Simplify your answer]
**Explanation:**
To solve these problems, we'll need to use the definition of conditional probability:
\[ P(C \mid A') = \frac{P(C \cap A')}{P(A')} \]
\[ P(C' \mid A') = \frac{P(C' \cap A')}{P(A')} \]
Start by calculating the individual probabilities and then use the formulas above to find the conditional probabilities.
---
*Note to educators: This text is designed to provide step-by-step instruction to help students understand how to calculate and simplify the conditional probabilities involving two events when rolling two fair six-sided dice.*
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