You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.7. With water it will die with probability 0.45. You are 83 % certain the neighbor will remember to water the plant. When you are on vacation, find the probability that the plant will die. Answer: You come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it? Answer:
You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.7. With water it will die with probability 0.45. You are 83 % certain the neighbor will remember to water the plant. When you are on vacation, find the probability that the plant will die. Answer: You come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it? Answer:
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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Question
![**Probability and Decision Making: Watering a Plant**
In this exercise, we explore the application of probability in decision-making scenarios. Consider the following situation:
**Scenario:**
You ask a neighbor to water a sickly plant while you are on vacation. Two critical probabilities concerning the plant’s well-being are provided:
1. Without water, the plant will die with a probability of 0.7.
2. With water, the plant will die with a probability of 0.45.
Additionally, you are 83% certain that the neighbor will remember to water the plant.
**Questions:**
1. When you are on vacation, what is the probability that the plant will die?
- **Answer:**
2. You come back from vacation and the plant is dead. What is the probability that the neighbor forgot to water it?
- **Answer:**
The problem requires applying the concepts of conditional probability to determine the answers. Calculating these probabilities involves understanding the given conditions and the relationship between them.
### Explanation and Calculation:
To solve these questions efficiently:
1. **Probability that the plant will die (P(D)) when you are on vacation:**
- Let's denote:
- \( P(D|W) \) as the probability the plant dies given it is watered = 0.45
- \( P(D|\neg W) \) as the probability the plant dies given it is not watered = 0.7
- \( P(W) \) as the probability that the neighbor waters the plant = 0.83
- \( P(\neg W) \) as the probability that the neighbor does not water the plant = 1 - 0.83 = 0.17
The total probability that the plant dies can be found using the law of total probability:
\[
P(D) = P(D|W)P(W) + P(D|\neg W)P(\neg W)
\]
Substituting the given values:
\[
P(D) = (0.45 \times 0.83) + (0.7 \times 0.17)
\]
2. **Probability that the neighbor forgot to water the plant (P(\neg W | D)) given that it is dead when you return:**
- This scenario involves using Bayes' Theorem:
\[
P(\neg W | D) = \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faed2933e-b67f-42a0-9fac-b6dce0e828b0%2F6938c468-17b5-43e2-ad24-85c688fc8c5c%2Foyfpok_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability and Decision Making: Watering a Plant**
In this exercise, we explore the application of probability in decision-making scenarios. Consider the following situation:
**Scenario:**
You ask a neighbor to water a sickly plant while you are on vacation. Two critical probabilities concerning the plant’s well-being are provided:
1. Without water, the plant will die with a probability of 0.7.
2. With water, the plant will die with a probability of 0.45.
Additionally, you are 83% certain that the neighbor will remember to water the plant.
**Questions:**
1. When you are on vacation, what is the probability that the plant will die?
- **Answer:**
2. You come back from vacation and the plant is dead. What is the probability that the neighbor forgot to water it?
- **Answer:**
The problem requires applying the concepts of conditional probability to determine the answers. Calculating these probabilities involves understanding the given conditions and the relationship between them.
### Explanation and Calculation:
To solve these questions efficiently:
1. **Probability that the plant will die (P(D)) when you are on vacation:**
- Let's denote:
- \( P(D|W) \) as the probability the plant dies given it is watered = 0.45
- \( P(D|\neg W) \) as the probability the plant dies given it is not watered = 0.7
- \( P(W) \) as the probability that the neighbor waters the plant = 0.83
- \( P(\neg W) \) as the probability that the neighbor does not water the plant = 1 - 0.83 = 0.17
The total probability that the plant dies can be found using the law of total probability:
\[
P(D) = P(D|W)P(W) + P(D|\neg W)P(\neg W)
\]
Substituting the given values:
\[
P(D) = (0.45 \times 0.83) + (0.7 \times 0.17)
\]
2. **Probability that the neighbor forgot to water the plant (P(\neg W | D)) given that it is dead when you return:**
- This scenario involves using Bayes' Theorem:
\[
P(\neg W | D) = \frac
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