Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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.
### 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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
