Question 1170727: 100,000 random people were tested for Condition A. Roberto's doctor told him he tested positive for Condition A. If the test is 99% accurate and Condition A is rare—one out of every thousand people have it.
What is the probability that testing positive means Roberto has Condition A?
A.99.0%
B.1%
C.9%
D.99.9%
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! This is a classic example of a problem that requires Bayes' Theorem to solve accurately. Here's how to break it down:
**1. Define Events**
* A: Roberto has Condition A.
* + : Roberto tests positive for Condition A.
**2. Given Probabilities**
* P(A) = 1/1000 = 0.001 (Prevalence of Condition A)
* P(¬A) = 999/1000 = 0.999 (Probability of not having Condition A)
* P(+ | A) = 0.99 (Probability of a positive test given Condition A)
* P(- | A) = 0.01 (Probability of a negative test given Condition A)
* P(+ | ¬A) = 0.01 (Probability of a positive test given no Condition A)
* P(- | ¬A) = 0.99 (Probability of a negative test given no Condition A)
**3. What We Want to Find**
We want to find P(A | +), the probability that Roberto has Condition A given that he tested positive.
**4. Bayes' Theorem**
Bayes' Theorem states:
P(A | +) = [P(+ | A) * P(A)] / P(+)
To find P(+), we use the law of total probability:
P(+) = P(+ | A) * P(A) + P(+ | ¬A) * P(¬A)
**5. Calculate P(+)**
P(+) = (0.99 * 0.001) + (0.01 * 0.999)
P(+) = 0.00099 + 0.00999
P(+) = 0.01098
**6. Calculate P(A | +)**
P(A | +) = (0.99 * 0.001) / 0.01098
P(A | +) = 0.00099 / 0.01098
P(A | +) ≈ 0.09016
**7. Convert to Percentage**
0.09016 * 100% ≈ 9.016%
**8. Choose the Closest Answer**
The closest answer to 9.016% is 9%.
**Therefore, the correct answer is C. 9%**
|
|
|
