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And the answer is 0.0333 or a 1/3 and now we're going to
apply the entire arsenal of inference we just learned about.
The joint probability of cancer and positive is 0.1 0.9. That's the joint that's not normalized.
So let's normalize it and we normalize it by the sum of the joint
for cancer and the joint for non-cancer.
Joint for cancer we just computed but the joint for non-cancer assumes the opposite prior 1-0.1
and it applies the positive result of a non-cancer case.
Now because the specificity first is negative, we have to do the
same trick as before and multiply it with 1-0.8.
When you worked this out, you find this to be 0 to 0.9 divided 0 to 0.9 + 0.9 0.2 that is 0.18
So if you put these all of this together, you get exactly a third.