On the basis of a physical examination, a doctor determines the probability of no tumour (event...

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On the basis of a physical examination, a doctor determines theprobability of no tumour   (event labelled C for‘clear’), a benign tumour (B) or a malignant tumour (M) as 0.7, 0.2and 0.1 respectively.
A further, in depth, test is conducted on the patient which canyield either a negative (N) result or positive (P). The test givesa negative result with probability 0.9 if no tumour is present(i.e. P(N|C) = 0.9). The test gives a negative result withprobability 0.8 if there is a benign tumour and 0.2 if there is amalignant tumour.
(i) Given this information calculatethe joint and marginal probabilities and display in the tablebelow.
Positive (P)
Negative (N)
MP
Clear          (C)
0.07
0.63
0.7
Benign        (B)
0.04
0.16
0.2
Malignant   (M)
0.08
0.02
0.1
MP
0.19
0.81
1
Obtain the posterior probability distribution for the patientwhen the test result is
           a) positive,   b) negative
Comment on how the test results change the doctor’s view of thepresence of a tumour.

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i Positive P Negative N Total Clear C 007 063 07 Benign B 004 016 02 Malignant M 008 002 01 Total 019 081 1 Given PC 07 PB 02 PM 01 PN C 09 PN and C PC 09 PN and C 09 PC 09 07 063 PP and C PC PN and C 07 063 007 PN B 08 PN and B PB 08 PN and B 08 PB 08 02 016 PP and B PB PN and B 02 016 004 See Answer

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	Positive (P)	Negative (N)	MP
Clear (C)	0.07	0.63	0.7
Benign (B)	0.04	0.16	0.2
Malignant (M)	0.08	0.02	0.1
MP	0.19	0.81	1