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Let's use the cancer example from my last unit.
There's a specific cancer that occurs in 1% of the population,
and a test for this cancer and with 90% chance
it is positive if they have this cancer, C.
That's usually called the sensitivity.
But the test sometimes is positive,
even if you don't have C.
Let's say with another 90% chance
it's negative if we don't have C.
That's usually called the specificity.
So here's my question.
Without further symptoms, you take the test,
and the test comes back positive.
What do you think is the probability of having
that specific type of cancer?
To answer this, let's draw a diagram.
Suppose these are all of the people,
and some of them, exactly 1%, have cancer.
99% is cancer free.
We know there's a test
that if you have cancer,
correctly diagnose it with 90% chance.
So if we draw the area where the test is positive,
cancer and test positive,
then this area over here
is 90% of the cancer circle.
However, this isn't the full truth.
The test sent out as positive
even if the person doesn't have cancer.
In fact, in our case,
it happened to be in 10% of all cases.
So we have to add more area,
because as big as 10% of this large area
is as big as 10% of this large area
where the test might go positive,
but the person doesn't have cancer.
So this blue area is 10% of all the area over here
minus the little small cancer circle.
And clearly, all the area outside these circles
corresponds a situation of no cancer,
and the test is negative.
So let me ask you again.
Suppose we have a positive test,
what do you think?
Would a prior probability of cancer
of 1%,
a sensitivity and specificity of 90%,
Do you think your new chances are now
90%
or 8%
or still just 1%?