Homework Help is Here – Start Your Trial Now!
Math
Probability
Not studied in library Total 55 60 Find the probability that a randomly chosen student will improve their grade given that they mave studied in the library?

Not studied in library Total 55 60 Find the probability that a randomly chosen student will improve their grade given that they mave studied in the library?

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...
See similar textbooks
icon
Related questions
Question

set it up first, then go back and simplify as much as possible. Ive also included some formulas that might help but im unsure.

## Key Formulas for Probability and Combinatorics ### Probability 1. **Joint Probability (Intersection of Events)** \[ P(A \text{ and } B) = P(A) \cdot P(B) \] 2. **Union of Events** \[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \] ### Permutations and Combinations 1. **Permutations** Permutations of \( n \) items taken \( r \) at a time: \[ nPr = \frac{n!}{(n-r)!} \] 2. **Combinations** Combinations of \( n \) items taken \( r \) at a time: \[ nCr = \frac{n!}{(n-r)!r!} \] ### Length Context - For \( L \): \[ L = 0.25n \] - For \( l \): \[ l = 0.75n \] ### Diagram Explanation This table presents fundamental formulas in probability and combinatorics that may be needed for solving particular problems. It includes formulas for the probability of the intersection and union of two events, as well as the formulas for calculating permutations and combinations of a given set of elements. - **Probability Formulas** provide the means to determine the likelihood of events occurring concurrently or individually. - **Permutation and Combination Formulas** are essential for calculating the number of ways to arrange or select items from a larger set where order does or does not matter. Additionally, length contexts with expressions \( L = 0.25n \) and \( l = 0.75n \) might refer to specific geometric or contextual values pertinent to the problem at hand.
Transcribed Image Text:## Key Formulas for Probability and Combinatorics ### Probability 1. **Joint Probability (Intersection of Events)** \[ P(A \text{ and } B) = P(A) \cdot P(B) \] 2. **Union of Events** \[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \] ### Permutations and Combinations 1. **Permutations** Permutations of \( n \) items taken \( r \) at a time: \[ nPr = \frac{n!}{(n-r)!} \] 2. **Combinations** Combinations of \( n \) items taken \( r \) at a time: \[ nCr = \frac{n!}{(n-r)!r!} \] ### Length Context - For \( L \): \[ L = 0.25n \] - For \( l \): \[ l = 0.75n \] ### Diagram Explanation This table presents fundamental formulas in probability and combinatorics that may be needed for solving particular problems. It includes formulas for the probability of the intersection and union of two events, as well as the formulas for calculating permutations and combinations of a given set of elements. - **Probability Formulas** provide the means to determine the likelihood of events occurring concurrently or individually. - **Permutation and Combination Formulas** are essential for calculating the number of ways to arrange or select items from a larger set where order does or does not matter. Additionally, length contexts with expressions \( L = 0.25n \) and \( l = 0.75n \) might refer to specific geometric or contextual values pertinent to the problem at hand.
**Probability and Statistics: Determining Conditional Probability from a Two-Way Table** The table below shows the results of 115 students who studied or did not study in the library, and whether their grades improved or did not improve. | | Improved Grade | Not Improved Grade | Total | |-------------------------|----------------|--------------------|-------| | Studied in the library | 45 | 5 | 50 | | Not studied in library | 10 | 55 | 65 | | **Total** | **55** | **60** | **115**| **Analysis:** - **Studied in the library:** 50 students - 45 students improved their grades. - 5 students did not improve their grades. - **Not studied in library:** 65 students - 10 students improved their grades. - 55 students did not improve their grades. **Calculating Conditional Probability:** We are tasked with finding the probability that a randomly chosen student will improve their grade given that they have studied in the library. This is a conditional probability, represented as \( P(\text{Improved Grade} \mid \text{Studied in the library}) \). The formula for conditional probability is: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] Where: - \( A \) is the event that the student's grade improved. - \( B \) is the event that the student studied in the library. - \( P(A \cap B) \) is the probability that both events A and B occur (studied in the library and improved grade). - \( P(B) \) is the probability that event B occurs (studied in the library). From the table: - Number of students who studied in the library and improved grades ( \( P(A \cap B) \) ) = 45. - Total number of students who studied in the library ( \( P(B) \) ) = 50. Thus, the conditional probability: \[ P(\text{Improved Grade} \mid \text{Studied in the library}) = \frac{45}{50} = 0.9 \] Therefore, the probability that a randomly chosen student will improve their grade given that they have studied in the library is \( 0.9 \) or 90%.
Transcribed Image Text:**Probability and Statistics: Determining Conditional Probability from a Two-Way Table** The table below shows the results of 115 students who studied or did not study in the library, and whether their grades improved or did not improve. | | Improved Grade | Not Improved Grade | Total | |-------------------------|----------------|--------------------|-------| | Studied in the library | 45 | 5 | 50 | | Not studied in library | 10 | 55 | 65 | | **Total** | **55** | **60** | **115**| **Analysis:** - **Studied in the library:** 50 students - 45 students improved their grades. - 5 students did not improve their grades. - **Not studied in library:** 65 students - 10 students improved their grades. - 55 students did not improve their grades. **Calculating Conditional Probability:** We are tasked with finding the probability that a randomly chosen student will improve their grade given that they have studied in the library. This is a conditional probability, represented as \( P(\text{Improved Grade} \mid \text{Studied in the library}) \). The formula for conditional probability is: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] Where: - \( A \) is the event that the student's grade improved. - \( B \) is the event that the student studied in the library. - \( P(A \cap B) \) is the probability that both events A and B occur (studied in the library and improved grade). - \( P(B) \) is the probability that event B occurs (studied in the library). From the table: - Number of students who studied in the library and improved grades ( \( P(A \cap B) \) ) = 45. - Total number of students who studied in the library ( \( P(B) \) ) = 50. Thus, the conditional probability: \[ P(\text{Improved Grade} \mid \text{Studied in the library}) = \frac{45}{50} = 0.9 \] Therefore, the probability that a randomly chosen student will improve their grade given that they have studied in the library is \( 0.9 \) or 90%.
Expert Solution
steps

Step by step

Solved in 3 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer