The probabilities of events A, B, and An B are given. Find (a) P(AUB), (b) the odds in favor of and the odds against A, (c) th against B, and (d) the odds in favor of and against A n B. 5 P(A) = —, P(B) = - P(B)=P(ANB)=0 11 11 (a) P(A U B) = 1 (Simplify your answer.) (b) The odds in favor of A are 5:6 (Simplify your answer Type whole numbers)
The probabilities of events A, B, and An B are given. Find (a) P(AUB), (b) the odds in favor of and the odds against A, (c) th against B, and (d) the odds in favor of and against A n B. 5 P(A) = —, P(B) = - P(B)=P(ANB)=0 11 11 (a) P(A U B) = 1 (Simplify your answer.) (b) The odds in favor of A are 5:6 (Simplify your answer Type whole numbers)
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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Part D
![### Probability and Odds in Events A and B
#### Introduction
The probabilities of events A, B, and A ∩ B are given. We are to find:
- **P(A ∪ B)**
- The odds in favor of and against A.
- The odds in favor of and against B.
- The odds in favor of and against A ∩ B.
**Given:**
\[ P(A) = \frac{5}{11}, \quad P(B) = \frac{6}{11}, \quad P(A \cap B) = 0 \]
#### Tasks
**(a) Find P(A ∪ B)**
\[ \text{P}(A \cup B) = 1 \quad \text{(Simplify your answer.)} \]
**(b) The Odds in Favor and Against A**
The odds in favor of A are:
\[ \frac{5}{6} \]
(Simplify your answer. Type whole numbers.)
The odds against A are:
\[ \frac{6}{5} \]
(Simplify your answer. Type whole numbers.)
**(c) The Odds in Favor and Against B**
The odds in favor of B are:
\[ \frac{6}{5} \]
(Simplify your answer. Type whole numbers.)
The odds against B are:
\[ \frac{5}{6} \]
(Simplify your answer. Type whole numbers.)
**(d) The Odds in Favor and Against A ∩ B**
The odds in favor of A ∩ B are:
\[ \frac{0}{11} \]
(Simplify your answer. Type whole numbers.)
The odds against A ∩ B are:
\[ \frac{11}{0} \]
(Simplify your answer. Type whole numbers.)
### Explanation
- **P(A ∪ B)**: We use the formula for the probability of the union of two events.
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since \( P(A \cap B) = 0 \):
\[ P(A \cup B) = \frac{5}{11} + \frac{6}{11} = 1 \]
- **Odds in Favor and Against A**:
- Odds in favor of A is the ratio of \( P(A) \) to \( P(A^c) \), where \( P(A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b1d7e88-5924-4190-9ef7-53bc83ebfef1%2Fca2dda9b-0a65-491e-81f3-0de7cbf717d1%2Fxjwhveu_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability and Odds in Events A and B
#### Introduction
The probabilities of events A, B, and A ∩ B are given. We are to find:
- **P(A ∪ B)**
- The odds in favor of and against A.
- The odds in favor of and against B.
- The odds in favor of and against A ∩ B.
**Given:**
\[ P(A) = \frac{5}{11}, \quad P(B) = \frac{6}{11}, \quad P(A \cap B) = 0 \]
#### Tasks
**(a) Find P(A ∪ B)**
\[ \text{P}(A \cup B) = 1 \quad \text{(Simplify your answer.)} \]
**(b) The Odds in Favor and Against A**
The odds in favor of A are:
\[ \frac{5}{6} \]
(Simplify your answer. Type whole numbers.)
The odds against A are:
\[ \frac{6}{5} \]
(Simplify your answer. Type whole numbers.)
**(c) The Odds in Favor and Against B**
The odds in favor of B are:
\[ \frac{6}{5} \]
(Simplify your answer. Type whole numbers.)
The odds against B are:
\[ \frac{5}{6} \]
(Simplify your answer. Type whole numbers.)
**(d) The Odds in Favor and Against A ∩ B**
The odds in favor of A ∩ B are:
\[ \frac{0}{11} \]
(Simplify your answer. Type whole numbers.)
The odds against A ∩ B are:
\[ \frac{11}{0} \]
(Simplify your answer. Type whole numbers.)
### Explanation
- **P(A ∪ B)**: We use the formula for the probability of the union of two events.
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since \( P(A \cap B) = 0 \):
\[ P(A \cup B) = \frac{5}{11} + \frac{6}{11} = 1 \]
- **Odds in Favor and Against A**:
- Odds in favor of A is the ratio of \( P(A) \) to \( P(A^c) \), where \( P(A
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