Urgent. i would really appreciate it!

According to the table,

A. By the law of total probability,

B. By definition of conditional probability,

That is, the test has an 80% probability of returning a true positive.

C. As in (B), we have by definition

but we don't yet know . By the law of total probability,

Then

That is, the probability that a patient has cancer given that the test returns a positive result is about 7.5%.

D. Simply put, the events and are not the same, and the difference is derived directly from the fact that .