Answer :
To tackle this problem, we need to determine three conditional probabilities based on the given two-way frequency table:
1. [tex]\( P(\text{Has disease} \mid \text{Test is negative}) \)[/tex]
2. [tex]\( P(\text{False positive} \mid \text{Doesn't have disease}) \)[/tex]
3. [tex]\( P(\text{Doesn't have disease} \mid \text{Test is positive}) \)[/tex]
We will analyze each probability step-by-step.
### 1. [tex]\( P(\text{Has disease} \mid \text{Test is negative}) \)[/tex]
This is the probability that a plant has the disease given that the test result is negative. Mathematically, it is expressed as:
[tex]\[ P(\text{Has disease} \mid \text{Test is negative}) = \frac{P(\text{Has disease and Test is negative})}{P(\text{Test is negative})} \][/tex]
From the table:
- The number of plants that have the disease and tested negative is [tex]\(0\)[/tex].
- The total number of plants that tested negative is [tex]\(943\)[/tex].
Thus,
[tex]\[ P(\text{Has disease} \mid \text{Test is negative}) = \frac{0}{943} = 0 \% \][/tex]
### 2. [tex]\( P(\text{False positive} \mid \text{Doesn't have disease}) \)[/tex]
This is the probability that the test gives a false positive result given that the plant doesn't have the disease. A false positive occurs when the test is positive but the plant doesn't have the disease. Mathematically, it is expressed as:
[tex]\[ P(\text{False positive} \mid \text{Doesn't have disease}) = \frac{P(\text{Doesn't have disease and Test is positive})}{P(\text{Doesn't have disease})} \][/tex]
From the table:
- The number of plants that don't have the disease and tested positive is [tex]\(7\)[/tex].
- The total number of plants that don't have the disease is [tex]\(950\)[/tex].
Thus,
[tex]\[ P(\text{False positive} \mid \text{Doesn't have disease}) = \frac{7}{950} \approx 0.7368 \% \][/tex]
### 3. [tex]\( P(\text{Doesn't have disease} \mid \text{Test is positive}) \)[/tex]
This is the probability that a plant doesn't have the disease given that the test result is positive. Mathematically, it is expressed as:
[tex]\[ P(\text{Doesn't have disease} \mid \text{Test is positive}) = \frac{P(\text{Doesn't have disease and Test is positive})}{P(\text{Test is positive})} \][/tex]
From the table:
- The number of plants that don't have the disease and tested positive is [tex]\(7\)[/tex].
- The total number of plants that tested positive is [tex]\(57\)[/tex].
Thus,
[tex]\[ P(\text{Doesn't have disease} \mid \text{Test is positive}) = \frac{7}{57} \approx 12.28 \% \][/tex]
### Summary
Based on the calculations:
- [tex]\( P(\text{Has disease} \mid \text{Test is negative}) = 0\% \)[/tex]
- [tex]\( P(\text{False positive} \mid \text{Doesn't have disease}) \approx 0.7368\% \)[/tex]
- [tex]\( P(\text{Doesn't have disease} \mid \text{Test is positive}) \approx 12.28\% \)[/tex]
Therefore, these are the correct values for the probabilities.
1. [tex]\( P(\text{Has disease} \mid \text{Test is negative}) \)[/tex]
2. [tex]\( P(\text{False positive} \mid \text{Doesn't have disease}) \)[/tex]
3. [tex]\( P(\text{Doesn't have disease} \mid \text{Test is positive}) \)[/tex]
We will analyze each probability step-by-step.
### 1. [tex]\( P(\text{Has disease} \mid \text{Test is negative}) \)[/tex]
This is the probability that a plant has the disease given that the test result is negative. Mathematically, it is expressed as:
[tex]\[ P(\text{Has disease} \mid \text{Test is negative}) = \frac{P(\text{Has disease and Test is negative})}{P(\text{Test is negative})} \][/tex]
From the table:
- The number of plants that have the disease and tested negative is [tex]\(0\)[/tex].
- The total number of plants that tested negative is [tex]\(943\)[/tex].
Thus,
[tex]\[ P(\text{Has disease} \mid \text{Test is negative}) = \frac{0}{943} = 0 \% \][/tex]
### 2. [tex]\( P(\text{False positive} \mid \text{Doesn't have disease}) \)[/tex]
This is the probability that the test gives a false positive result given that the plant doesn't have the disease. A false positive occurs when the test is positive but the plant doesn't have the disease. Mathematically, it is expressed as:
[tex]\[ P(\text{False positive} \mid \text{Doesn't have disease}) = \frac{P(\text{Doesn't have disease and Test is positive})}{P(\text{Doesn't have disease})} \][/tex]
From the table:
- The number of plants that don't have the disease and tested positive is [tex]\(7\)[/tex].
- The total number of plants that don't have the disease is [tex]\(950\)[/tex].
Thus,
[tex]\[ P(\text{False positive} \mid \text{Doesn't have disease}) = \frac{7}{950} \approx 0.7368 \% \][/tex]
### 3. [tex]\( P(\text{Doesn't have disease} \mid \text{Test is positive}) \)[/tex]
This is the probability that a plant doesn't have the disease given that the test result is positive. Mathematically, it is expressed as:
[tex]\[ P(\text{Doesn't have disease} \mid \text{Test is positive}) = \frac{P(\text{Doesn't have disease and Test is positive})}{P(\text{Test is positive})} \][/tex]
From the table:
- The number of plants that don't have the disease and tested positive is [tex]\(7\)[/tex].
- The total number of plants that tested positive is [tex]\(57\)[/tex].
Thus,
[tex]\[ P(\text{Doesn't have disease} \mid \text{Test is positive}) = \frac{7}{57} \approx 12.28 \% \][/tex]
### Summary
Based on the calculations:
- [tex]\( P(\text{Has disease} \mid \text{Test is negative}) = 0\% \)[/tex]
- [tex]\( P(\text{False positive} \mid \text{Doesn't have disease}) \approx 0.7368\% \)[/tex]
- [tex]\( P(\text{Doesn't have disease} \mid \text{Test is positive}) \approx 12.28\% \)[/tex]
Therefore, these are the correct values for the probabilities.
