Question4Max. score: 2.00A test has a true positive rate of 100% and false positive rate of 5%. There is a population with a 1/1000 rate of having the condition the test identifies(only 1 person in 1000 actually has the condition). Considering a positive test, what is the probability of having that condition?2%3%3.5%3.2%
Question
Question4Max. score: 2.00A test has a true positive rate of 100% and false positive rate of 5%. There is a population with a 1/1000 rate of having the condition the test identifies(only 1 person in 1000 actually has the condition). Considering a positive test, what is the probability of having that condition?2%3%3.5%3.2%
Solution
To calculate the probability of having the condition given a positive test, we can use Bayes' theorem.
Let's denote the following: P(C) = probability of having the condition = 1/1000 P(Pos|C) = true positive rate = 100% = 1 P(Pos|¬C) = false positive rate = 5% = 0.05
We want to find P(C|Pos), the probability of having the condition given a positive test.
According to Bayes' theorem: P(C|Pos) = (P(Pos|C) * P(C)) / P(Pos)
To calculate P(Pos), we need to consider both true positive and false positive cases: P(Pos) = P(Pos|C) * P(C) + P(Pos|¬C) * P(¬C)
Since P(¬C) = 1 - P(C), we can substitute it into the equation: P(Pos) = P(Pos|C) * P(C) + P(Pos|¬C) * (1 - P(C))
Now we can substitute the given values into the equation: P(Pos) = 1 * (1/1000) + 0.05 * (1 - 1/1000)
Simplifying the equation: P(Pos) = 0.001 + 0.05 - 0.00005 P(Pos) = 0.05095
Finally, we can substitute P(Pos) back into Bayes' theorem to find P(C|Pos): P(C|Pos) = (P(Pos|C) * P(C)) / P(Pos) P(C|Pos) = (1 * (1/1000)) / 0.05095
Calculating the value: P(C|Pos) ≈ 0.0196
Therefore, the probability of having the condition given a positive test is approximately 1.96%, which is closest to the option 2% provided in the question.
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