Question

9. Suppose a certain type of cancer is diagnosed by first administering a test, looking at the test result and then making the diagnosis. If the test is positive, the patient is diagnosed as having the cancer. If the test is negative, the person is diagnosed as not having the cancer. Let event the person has the cancer and event P the test result is positive. Being diagnosed as having (not having) the cancer does not mean the person has (does not have) the cancer. (a) List all four outcomes (hint: think of this as a two stage process.) (b) Would you assign probability 1/4 to each of the outcomes? Explain why or why not. (3.3.14)

Answer 1

Answer

Here by the problem

A certain type of cancer is diagnosed by first administering a test, looking at the test result then making a diagnosis. If the test is positive, the patient is diagnosed as having the cancer. If the test is negative, the person is diagnosed as not having the cancer.

Here we denote two event P and C as

C=Event that the person has the cancer

P=Event that the result is positive.

(a) Here all the possible four outcomes be,

$(CP,C^cP,CP^c,C^cP^c)$ where CP implies $C\cup P$ or the event that the patient has cancer and the test results as positive.

(b) If we assign probability 1/4 to each of the outcome , that is $P(CP)=P(C^cP)=P(CP^c)=P(C^cP^c)=1/4$ then that implies,

$P(C)=P(CP)+P(CP^c)=1/2$ and similarly $P(P)=1/2$

So then we can see

$P(CP)=1/4=1/2*1/2=P(C).P(P)$

An it is valid for other events as well.

So C and P are independent.

But logically it is very unlikely to happen. As the test result will very usually show positive for the person who is actually having cancer implying sort of dependence between C and P

SO it is preferable not to assign such values.

Hence the answer................

Thank you...............