Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help. a. Consider the events: E = {the card is an even-numbered card}, F = {The card is an odd numbered card}. Are these events independent, mutually exclusive, both or neither? Justify.
Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help. a. Consider the events: E = {the card is an even-numbered card}, F = {The card is an odd numbered card}. Are these events independent, mutually exclusive, both or neither? Justify.
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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2. Two dice are rolled. Find the
a. Consider the
![**Probability and Statistics Concepts**
**Basic Probability Principle**
\[ P(E) = \frac{n(E)}{n(S)} \]
Where \( n(E) \) is the number of favorable outcomes, and \( n(S) \) is the total number of possible outcomes.
**Union Rule**
- For sets: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
- For probability: \( P(E \cup F) = P(E) + P(F) - P(E \cap F) \)
**Product Rule**
\[ P(E \cap F) = P(E) \cdot P(F|E) = P(F) \cdot P(E|F) \]
**Complement Rule (Probability)**
- \( P(E') = 1 - P(E) \)
- \( P(E) = 1 - P(E') \)
**Complement Rule (Number of Outcomes)**
- \( n(E) = n(S) - n(E') \)
- \( n(E') = n(S) - n(E) \)
**Conditional Probability**
\[ P(E|F) = \frac{n(E \cap F)}{n(F)} = \frac{P(E \cap F)}{P(F)} \]
**Number of Subsets**
If a set has \( n \) elements, it has \( 2^n \) subsets.
**Odds**
- Odds in favor of \( E \): \( \frac{P(E)}{P(E')} \)
- If odds in favor of \( E \) are \( m : n \), then \( P(E) = \frac{m}{m+n} \)
**Bayes’ Theorem**
\[ P(F|E) = \frac{P(E|F) \cdot P(F)}{P(E)} \]
- Special Case:
\[ P(F|E) = \frac{P(F) \cdot P(E|F)}{P(F) \cdot P(E|F) + P(F') \cdot P(E|F')} \]
**Permutations**
\[ P(n, k) = \frac{n!}{(n-k)!} \]
**Combinations**
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
**Distinguishable Permutations**
\[ \frac{n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F985fbbf0-f8fa-4c87-a1a5-f926559935c9%2Fdd1799e1-001e-4165-adc0-3cc53c6d866e%2Fyo75c5l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability and Statistics Concepts**
**Basic Probability Principle**
\[ P(E) = \frac{n(E)}{n(S)} \]
Where \( n(E) \) is the number of favorable outcomes, and \( n(S) \) is the total number of possible outcomes.
**Union Rule**
- For sets: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
- For probability: \( P(E \cup F) = P(E) + P(F) - P(E \cap F) \)
**Product Rule**
\[ P(E \cap F) = P(E) \cdot P(F|E) = P(F) \cdot P(E|F) \]
**Complement Rule (Probability)**
- \( P(E') = 1 - P(E) \)
- \( P(E) = 1 - P(E') \)
**Complement Rule (Number of Outcomes)**
- \( n(E) = n(S) - n(E') \)
- \( n(E') = n(S) - n(E) \)
**Conditional Probability**
\[ P(E|F) = \frac{n(E \cap F)}{n(F)} = \frac{P(E \cap F)}{P(F)} \]
**Number of Subsets**
If a set has \( n \) elements, it has \( 2^n \) subsets.
**Odds**
- Odds in favor of \( E \): \( \frac{P(E)}{P(E')} \)
- If odds in favor of \( E \) are \( m : n \), then \( P(E) = \frac{m}{m+n} \)
**Bayes’ Theorem**
\[ P(F|E) = \frac{P(E|F) \cdot P(F)}{P(E)} \]
- Special Case:
\[ P(F|E) = \frac{P(F) \cdot P(E|F)}{P(F) \cdot P(E|F) + P(F') \cdot P(E|F')} \]
**Permutations**
\[ P(n, k) = \frac{n!}{(n-k)!} \]
**Combinations**
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
**Distinguishable Permutations**
\[ \frac{n
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