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I have a question about the false positive paradox. It is an example that is often used to motivate Baye's formula. I first give an explanation of the paradox.

Suppose there is a device which is 90 % accurate when screening for cancer. This means that $P(\text{positive result|cancer})=0.9$, and $P(\text{negative result|not cancer})=0.9$. Assume also that the chances of having cancer is $P(\text{cancer})=0.01$. Then we have $P(\text{cancer|positive result})=0.9\cdot0.01/(0.9\cdot0.01+0.99\cdot0.1)=0.083$.

The paradox is that even though the test seems to be very good, if you get a positive result, there is still high probability you don't have cancer.

My question is the definition of accuary of the test. Why is accuracy of this test defined this way? Let's say measure the accuary this way: You have a group of people, and you use your device to screen the members in your group for cancer. You then take out all those who got a positive result, and you do further work to see how many really had cancer. If you do this then $P(\text{cancer| positive result})$ would be the measure you used for accuary. Is this just plain wrong? Why isn't it done this way?

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    $\begingroup$ $P(\text{test pos}|\text{disease})$ is not the accuracy, it is called the sensitivity or true positive rate. $P(\text{disease} | \text{test pos})$ is not the accuracy either, it is the positive predictive value. This confusion has cast serious doubt on the weight of DNA testing in forensic evidence, as an example. $\endgroup$ – AdamO Nov 9 '17 at 21:43
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Definitions aren't really "right" or "wrong" - they are agreed upon. Your definition doesn't match what most people mean when they say accuracy; I think most people want "accuracy" to include both positive and negative results. There's no one good measure of a test; that's why we have sensitivity, specificity, false positive rate, false negative rate and so on (that is by no means all the measures).

Using your definition, we could make a test extremely accurate by simply saying almost no one had cancer. This is like having very high sensitivity but low specificity.

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Note that your suggested definition of "accuracy" is ambiguous. A test has at least two possible outcomes, and we'll get a different number depending on which of those outcomes we examine: P(cancer|positive result) will usually not be equal to P(not cancer|negative result), and it's not always obvious which of those should be the basis for "accuracy". In cases like cancer testing, false positives are sometimes just as devastating as false negatives.

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