Certainly! Let's carefully walk through the solution step-by-step using probability concepts.
### Given Information:
1. Positive Cases:
- Virus positive: 425
- No virus positive: 7960
2. Negative Cases:
- Virus negative: 75
- No virus negative: 91540
3. Totals:
- Total positive: 425 (virus positive) + 7960 (no virus positive) = 8385
- Total negative: 75 (virus negative) + 91540 (no virus negative) = 91615
- Total number of people: 100000
### Part (a): Finding [tex]\( P(A \mid B) \)[/tex]
This is the probability that a person has the virus given that they have tested positive.
Using the definition of conditional probability:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Where:
- [tex]\( P(A \cap B) \)[/tex] is the probability of testing positive and having the virus.
- [tex]\( P(B) \)[/tex] is the probability of testing positive.
From the data:
- [tex]\( P(A \cap B) = \frac{425}{100000} \)[/tex]
- [tex]\( P(B) = \frac{8385}{100000} \)[/tex]
So,
[tex]\[ P(A \mid B) = \frac{\frac{425}{100000}}{\frac{8385}{100000}} = \frac{425}{8385} \][/tex]
Now, calculate [tex]\(\frac{425}{8385} \)[/tex]:
[tex]\[ P(A \mid B) \approx 0.0507 \][/tex]
Therefore,
[tex]\[ P(A \mid B) = 5.07\% \][/tex]
### Part (b): Finding [tex]\( P(\neg A \mid \neg B) \)[/tex]
This is the probability that a person does not have the virus given that they test negative.
Using the definition of conditional probability:
[tex]\[ P(\neg A \mid \neg B) = \frac{P(\neg A \cap \neg B)}{P(\neg B)} \][/tex]
Where:
- [tex]\( P(\neg A \cap \neg B) \)[/tex] is the probability of testing negative and not having the virus.
- [tex]\( P(\neg B) \)[/tex] is the probability of testing negative.
From the data:
- [tex]\( P(\neg A \cap \neg B) = \frac{91540}{100000} \)[/tex]
- [tex]\( P(\neg B) = \frac{91615}{100000} \)[/tex]
So,
[tex]\[ P(\neg A \mid \neg B) = \frac{\frac{91540}{100000}}{\frac{91615}{100000}} = \frac{91540}{91615} \][/tex]
Now, calculate [tex]\(\frac{91540}{91615} \)[/tex]:
[tex]\[ P(\neg A \mid \neg B) \approx 0.9992 \][/tex]
Therefore,
[tex]\[ P(\neg A \mid \neg B) = 99.92\% \][/tex]
### Final Answers:
a. [tex]\( P(A \mid B) = 5.07 \)[/tex]
b. [tex]\( P(\neg A \mid \neg B) = 99.92 \)[/tex]
These results reflect the respective probabilities rounded to the nearest hundredth of a percent as required.
