Question
A certain medical test for a rare disease has an accuracy of 90% while the disease occurs in 1% of the population. If you take this test and the result comes back positive, what is the probability that you actually have the disease and what is the probability that the result is a false positive?
A.
0.9% chance of a true positive and 9.9% of a false positive
B.
1.0% chance of a true positiveand 99.0% of a false positive
C.
10.0% chance of a true positive and 90.0% of a false positive
D.
90.0% chance of a true positive and 10.0% of a false positive
A.
0.9% chance of a true positive and 9.9% of a false positive
B.
1.0% chance of a true positiveand 99.0% of a false positive
C.
10.0% chance of a true positive and 90.0% of a false positive
D.
90.0% chance of a true positive and 10.0% of a false positive
Answers
GPT-4o mini
To solve this problem, we can use Bayes' theorem. Let's define the following variables:
- Let \( D \) be the event that a person has the disease.
- Let \( T \) be the event that the test result is positive.
We know from the information given:
- \( P(D) = 0.01 \) (the prevalence of the disease in the population).
- \( P(T | D) = 0.90 \) (the probability of a positive test given the person has the disease, i.e., true positive rate).
- \( P(T | \neg D) = 0.10 \) (the probability of a positive test given the person does not have the disease, i.e., false positive rate, since accuracy is 90%).
We want to find \( P(D | T) \), the probability that a person has the disease given that they tested positive.
Using Bayes' theorem:
\[
P(D | T) = \frac{P(T | D) \cdot P(D)}{P(T)}
\]
Next, we need to calculate \( P(T) \), the total probability of a positive test result:
\[
P(T) = P(T | D) \cdot P(D) + P(T | \neg D) \cdot P(\neg D)
\]
We have:
- \( P(\neg D) = 1 - P(D) = 0.99 \)
Substituting in the values:
\[
P(T) = (0.90 \cdot 0.01) + (0.10 \cdot 0.99)
\]
\[
P(T) = 0.009 + 0.099 = 0.108
\]
Now we can find \( P(D | T) \):
\[
P(D | T) = \frac{0.90 \cdot 0.01}{0.108} = \frac{0.009}{0.108} \approx 0.0833 \, (\text{or } 8.33\%)
\]
Now, to find the probability of a false positive, we will compute \( P(\neg D | T) \):
\[
P(\neg D | T) = 1 - P(D | T)
\]
\[
P(\neg D | T) \approx 1 - 0.0833 = 0.9167 \, (\text{or about } 91.67\%)
\]
Now we can summarize our results as:
- \( P(D | T) \approx 8.33\% \) for a true positive.
- \( P(\neg D | T) \approx 91.67\% \) for a false positive.
Now none of the options provided in A, B, C, or D about the probabilities match this calculation.
Please double-check the options given or provide additional context to ensure we have interpreted the problem correctly.
- Let \( D \) be the event that a person has the disease.
- Let \( T \) be the event that the test result is positive.
We know from the information given:
- \( P(D) = 0.01 \) (the prevalence of the disease in the population).
- \( P(T | D) = 0.90 \) (the probability of a positive test given the person has the disease, i.e., true positive rate).
- \( P(T | \neg D) = 0.10 \) (the probability of a positive test given the person does not have the disease, i.e., false positive rate, since accuracy is 90%).
We want to find \( P(D | T) \), the probability that a person has the disease given that they tested positive.
Using Bayes' theorem:
\[
P(D | T) = \frac{P(T | D) \cdot P(D)}{P(T)}
\]
Next, we need to calculate \( P(T) \), the total probability of a positive test result:
\[
P(T) = P(T | D) \cdot P(D) + P(T | \neg D) \cdot P(\neg D)
\]
We have:
- \( P(\neg D) = 1 - P(D) = 0.99 \)
Substituting in the values:
\[
P(T) = (0.90 \cdot 0.01) + (0.10 \cdot 0.99)
\]
\[
P(T) = 0.009 + 0.099 = 0.108
\]
Now we can find \( P(D | T) \):
\[
P(D | T) = \frac{0.90 \cdot 0.01}{0.108} = \frac{0.009}{0.108} \approx 0.0833 \, (\text{or } 8.33\%)
\]
Now, to find the probability of a false positive, we will compute \( P(\neg D | T) \):
\[
P(\neg D | T) = 1 - P(D | T)
\]
\[
P(\neg D | T) \approx 1 - 0.0833 = 0.9167 \, (\text{or about } 91.67\%)
\]
Now we can summarize our results as:
- \( P(D | T) \approx 8.33\% \) for a true positive.
- \( P(\neg D | T) \approx 91.67\% \) for a false positive.
Now none of the options provided in A, B, C, or D about the probabilities match this calculation.
Please double-check the options given or provide additional context to ensure we have interpreted the problem correctly.