Question

On the basis of a physical examination, a doctor determines the probability of no tumour (event labelled C for ‘clear’), a benign tumour (B) or a malignant tumour (M) as 0.7, 0.2 and 0.1 respectively. A further, in depth, test is conducted on the patient which can yield either a negative (N) result or positive (P). The test gives a negative result with probability 0.9 if no tumour is present (i.e. P(N|C) = 0.9). The test gives a negative result with probability 0.8 if there is a benign tumour and 0.2 if there is a malignant tumour. (i) Given this information calculate the joint and marginal probabilities and display in the table below.

	Positive (P)	Negative (N)
Clear          (C)
Benign        (B)
Malignant   (M)
			1

(ii) What is the marginal probability the test result will be negative?

(iii) Obtain the posterior probability distribution for the patient when the test result is a) positive, b) negative

(iv) Comment on how the test results change the doctor’s view of the presence of a tumour.

Answer 1

i)

	P	N	total
C	0.07	0.63	0.7
B	0.04	0.16	0.2
M	0.08	0.02	0.1
total	0.19	0.81	1

ii) marginal probability the test result will be negative =0.81

iii)

posterior probability distribution for the patient when the test result is

a) positive is as follows:

C	0.3684
B	0.2105
M	0.4211

b) negative is as follows:

C	0.7778
B	0.1975
M	0.0247

iv)

as we can see that even coming a test to be positive does not give a higher probability of having a tumor therefore a retest is required to get higher reliability to operate.