Ten thousand screened

Question

Out of 10,000 people screened for a condition, 100 truly have it, and the test flags all 100 of them. Of the 9,900 people who do not have the condition, the test also flags 300 of them.

If a randomly chosen screened person tested positive, what is the probability they truly have the condition?

Accepted Answer

Everything is already counted for you, so just compare the two kinds of positives. - True positives (people who have it and are flagged): 100. - False positives (healthy people who are flagged): 300. Total positives: 100 + 300 = 400. The fraction of positives that are genuine is P(	ext{has it} \mid 	ext{positive}) = \frac{100}{400} = \frac{1}{4} = 25\%. Even though the test catches every real case, three out of four positive results are false alarms, because the healthy group is so much larger that its small error rate still produces plenty of positives. Takeaway. A positive result is only …

Ten thousand screened

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