I need help verifying if my reasoning is right. This is the example.
"The probability that a male develops some form of cancer in his lifetime is 0.4567. The probability that a male has at least one false-positive test result (meaning the test comes back for cancer when the man does not have it) is 0.51. Some of the following questions do not have enough information for you to answer them. Write “not enough information” for those answers. Let C = a man develops cancer in his lifetime and P = man has at least one false positive."
- $P(C)=0.4567$
- $P(P|C)=$ not enough information
- $P(P|C')=$ not enough information
- If a test comes up positive, based upon numerical values, can you assume that man has cancer? Justify numerically and explain why or why not. -> No, because over half (0.51) of men have at least one false-positive test.
On the book appears these as solutions and there is the tree diagram for this same problem in another exercise. This is the tree diagram.
Ok. My question is about items 2 and 3. I'm looking for the why there is not enough information.
2.
In exercise 2 there is not enough information because in order to find $P(P|C)$ I have to use the formula for conditional probability which is:
$$P(P|C)=\frac{P(P⋂C)}{P(C)}$$ $$P(P|C)=\frac{P(P⋂C)}{0.4567}$$
There is not enough information because the problem doesn't provide $P(P⋂C)$?
3.
For item 3 happens the same? The problem does not provide $P(P⋂C')$?
$$P(P|C')=\frac{P(P⋂C')}{P(C')}$$ $$P(P|C')=\frac{P(P⋂C')}{0.5433}$$