2. Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help. a. A single card is picked from a standard deck of 52 cards. Find the following probabilities. P(king I card is a heart) Are the events "king" and "heart" independent?
2. Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help. a. A single card is picked from a standard deck of 52 cards. Find the following probabilities. P(king I card is a heart) Are the events "king" and "heart" independent?
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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Question
2. Two dice are rolled. Find the probabilities of the following, as reduced fractions, and justify. Use the formula sheet if needed for help.
a. A single card is picked from a standard deck of 52 cards. Find the following probabilities. P(king I card is a heart) Are the
![**Basic Probability Principle**
\[
P(E) = \frac{n(E)}{n(S)}
\]
**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, then 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**
\[
\frac{n!}{n_1!n_2!\ldots n_k!}
\]
**Binomial Probability**
\[
P(k \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F985fbbf0-f8fa-4c87-a1a5-f926559935c9%2F64822b74-8314-43e1-b58f-65aaa1ac6841%2F7pw85vo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Basic Probability Principle**
\[
P(E) = \frac{n(E)}{n(S)}
\]
**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, then 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**
\[
\frac{n!}{n_1!n_2!\ldots n_k!}
\]
**Binomial Probability**
\[
P(k \
Expert Solution
Step 1
Given that, Two dice are rolled. We need to find:
P(king I card is a heart)
Step by step
Solved in 2 steps with 1 images
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