Different ways to calculate conditional probability of independent events

Question

Say we have TWO tests for cancer. Each test has the same probabilities of being right/wrong for cancer/no cancer. What is the probability of getting two positive test results?

In what follows, I'll use C for "has cancer", "+" for a positive test result, "++" for two positives (I don't distinguish between the tests because they have the same probabilities).

A. Use the formula for total probability:

P(++) = P(++|C) $\times$ P(C) $+$ P(++|$\neg$C) $\times$ P($\neg$C)

Since each test is independent of the other, apply the formula for joint probability:

= P(+|C)$^2$ $\times$ P(C) $+$ P(+|$\neg$C)$^2$ $\times$ P($\neg$C)

B. Since each test is independent of the other one, the joint probability is the product of the probability of each (which is the same):

P(++) = P(+)$^2$

Then, apply the formula for total probability to each:

= [P(+|C) $\times$ P(C) $+$ P(+|$\neg$C) $\times$ P($\neg$C)]$^2$

Obviously, the two are not equal. Why not? And which one is correct?

Answer 1

Jason's answer is certainly true, but I don't think it fully explains the reason your formulas don't match up, but only why you wouldn't want to apply this logic in the specific case of cancer tests.

Instead of + and ++, let's generalize this to events $A,B,C$. Your second calculation assumes $P(AB)=P(A)P(B)$, i.e. that $A$ and $B$ are marginally independent, which is in agreement with your assumption (although that assumption would not hold for cancer tests).

Your first calculation, however, assumes $P(AB|C) = P(A|C) P(B|C)$, i.e. that $A$ and $B$ are conditionally independent given $C$. These are NOT the same and in general one does not imply the other. This is easy to see in the Gaussian case: Gaussian r.v.s are independent if and only if their correlation is 0, but they are conditionally independent given another if and only if their partial correlation is 0. Setting up a correlation matrix with some zeros and inverting it to find the partial correlation matrix is a good way to see how these two quantities relate and that they are NOT interchangeable.

Answer 2

The two tests are not absolutely independent. They are conditionally independent on C.

Thinking about it intuitively, if you test positive for the first test, we can predict that it's more likely you'll test positive for the second test, hence the two are not absolutely independent.

However, given that you know that you have cancer, the two tests are independent i.e. knowing the outcome of the first will not affect our prediction of the outcome of the second. Hence, "conditional independence".