In a certain population, the probability that a randomly selected subject will have been exposed to a certain allergen and experience a reaction to the allergen is .60. The probabil- ity is .8 that a subject exposed to the allergen will experience an allergic reaction. If a subject is selected at random from this population, what is the probability that he or she will have been exposed to the allergen?
In a certain population, the probability that a randomly selected subject will have been exposed to a certain allergen and experience a reaction to the allergen is .60. The probabil- ity is .8 that a subject exposed to the allergen will experience an allergic reaction. If a subject is selected at random from this population, what is the probability that he or she will have been exposed to the allergen?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Probability and Conditional Probability Example**
Given Problem:
10. Do the problem #9 on page 78 in the Chapter 3 Review questions and exercises starting on page 76.
**Definitions and Given Information:**
- \(E = \text{Exposed}\)
- \(R = \text{Reaction}\)
\[ P(E \cap R) = P(\text{Exposed and Reaction}) = 0.60 \]
\[ P(R \mid E) = P(\text{Reaction given Exposed}) = 0.80 \]
**Objective:**
Find \( P(E) = P(\text{Exposed}) \)
**Formula for Conditional Probability:**
\[ P(R \mid E) = \frac{P(R \cap E)}{P(E)} \]
**Substitute Given Values:**
\[ P(0.80) = \frac{P(0.60)}{X} \]
\[ \frac{P(0.80)}{1} = \frac{P(0.60)}{X} \]
**Solving for X:**
\[ \frac{0.80X}{0.80} = \frac{0.60}{0.80} \]
\[ X = 0.75 \]
**Conclusion:**
The probability that he or she will have been exposed to the allergen is \( P(E) = 0.75 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc805eb-60ed-457b-ab8e-02c516698c74%2Fb20be556-b27b-4198-813d-de0d0bc097f4%2Fow35j98_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability and Conditional Probability Example**
Given Problem:
10. Do the problem #9 on page 78 in the Chapter 3 Review questions and exercises starting on page 76.
**Definitions and Given Information:**
- \(E = \text{Exposed}\)
- \(R = \text{Reaction}\)
\[ P(E \cap R) = P(\text{Exposed and Reaction}) = 0.60 \]
\[ P(R \mid E) = P(\text{Reaction given Exposed}) = 0.80 \]
**Objective:**
Find \( P(E) = P(\text{Exposed}) \)
**Formula for Conditional Probability:**
\[ P(R \mid E) = \frac{P(R \cap E)}{P(E)} \]
**Substitute Given Values:**
\[ P(0.80) = \frac{P(0.60)}{X} \]
\[ \frac{P(0.80)}{1} = \frac{P(0.60)}{X} \]
**Solving for X:**
\[ \frac{0.80X}{0.80} = \frac{0.60}{0.80} \]
\[ X = 0.75 \]
**Conclusion:**
The probability that he or she will have been exposed to the allergen is \( P(E) = 0.75 \).
![### Probability and Allergens
**Problem Statement:**
In a certain population, the probability that a randomly selected subject will have been exposed to a certain allergen and experience a reaction to the allergen is 0.60. The probability is 0.8 that a subject exposed to the allergen will experience an allergic reaction. If a subject is selected at random from this population, what is the probability that he or she will have been exposed to the allergen?
**Explanation:**
This problem involves conditional probability and can be visualized in terms of the following probabilities:
1. The joint probability of being exposed to an allergen and experiencing a reaction, P(E ∩ R) = 0.60.
2. The conditional probability of experiencing a reaction given exposure to the allergen, P(R|E) = 0.80.
We need to calculate the probability that a subject has been exposed to the allergen, P(E).
From the definition of conditional probability:
\[ P(R|E) = \frac{P(E ∩ R)}{P(E)} \]
We can rearrange this to solve for P(E):
\[ P(E) = \frac{P(E ∩ R)}{P(R|E)} \]
Substituting the given values:
\[ P(E) = \frac{0.60}{0.80} = 0.75 \]
So, the probability that a randomly selected subject from this population will have been exposed to the allergen is **0.75 or 75%**.
**Applications:**
This type of probability problem is crucial in fields such as epidemiology, public health, and risk assessment. It helps in understanding the spread and impact of allergens and other factors within a population and aids in making informed decisions for health management and policies.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc805eb-60ed-457b-ab8e-02c516698c74%2Fb20be556-b27b-4198-813d-de0d0bc097f4%2Ffiamjqf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability and Allergens
**Problem Statement:**
In a certain population, the probability that a randomly selected subject will have been exposed to a certain allergen and experience a reaction to the allergen is 0.60. The probability is 0.8 that a subject exposed to the allergen will experience an allergic reaction. If a subject is selected at random from this population, what is the probability that he or she will have been exposed to the allergen?
**Explanation:**
This problem involves conditional probability and can be visualized in terms of the following probabilities:
1. The joint probability of being exposed to an allergen and experiencing a reaction, P(E ∩ R) = 0.60.
2. The conditional probability of experiencing a reaction given exposure to the allergen, P(R|E) = 0.80.
We need to calculate the probability that a subject has been exposed to the allergen, P(E).
From the definition of conditional probability:
\[ P(R|E) = \frac{P(E ∩ R)}{P(E)} \]
We can rearrange this to solve for P(E):
\[ P(E) = \frac{P(E ∩ R)}{P(R|E)} \]
Substituting the given values:
\[ P(E) = \frac{0.60}{0.80} = 0.75 \]
So, the probability that a randomly selected subject from this population will have been exposed to the allergen is **0.75 or 75%**.
**Applications:**
This type of probability problem is crucial in fields such as epidemiology, public health, and risk assessment. It helps in understanding the spread and impact of allergens and other factors within a population and aids in making informed decisions for health management and policies.
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