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3. The occurrence of a certain cancer in the general population is 1 in 5,000. A diagnostic test for this cancer exists. If a patient has the cancer, the test is positive 95% of the time. If the patient does not have cancer, the test is negative 98% of the time. (a) Construct a tree diagram. Write probabilities on the branches for each of the two stages. Let the first stage consist of two events: Cancer, No Cancer. (b) What is the probability of having the cancer and the result being positive? Answer: 1.900-10-4 (c) Of keen interest to a patient is the probability of having the cancer given that the test turns out negative. Find this probability. Answer: 0.0000102 (d) Perhaps of greater importance to a patient is the probability of having the cancer given that the test turns out positive. Find this probability. Answer: 0.00941 (3.5.6)

math   Statistics-And-Probability   
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Answer #1

(a)

Let C shows the event that person has cancer. Let P shows the event that test gives positive result and N shows the negative result.

P(PIC) -0.95 P(C) 0.0002 0.05 P(P C) 0.02 P(C) -0.9998 - 0.98

(b)

P(P C) = P(PIC)P(C) = 0.95 . 0.0002 0.000 19

(c)

P(NIC)P(C) 0.05 0.0002 PlŅIC)P(C) P(NIC) P(C) 0.05 . 0.0002 0.98 , 0.9998 0.00000102

(d)

P PCP(C) 0.00019 P(C|P) 0.00941 Pl PC)P(C) + P(PIC) P(C) 0.000 19 0.02 . 0.9998

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