Lets say C' = Event male patient does not have prostate cancer.
Given that ,
P(+ | C') = 0.75
P(- | C)= 0.2
P(C) = 0.02
=> P(- | C')=0.25, P(+ | C)= 0.8,P(C')= 0.98
a)Probability that the male patient has prostate cancer if the PSA test comes back positive = P(C | +)
As per bayes theorem ,
P(C | +) =
=
=0.0213
b)
Probability that the male patient has prostate cancer if the PSA test comes back negative= P(C | -)
=
=
=0.0161
c)
If P(C) = 0.3
=>P(C')= 0.7
Probability that the male patient has prostate cancer if the PSA test comes back positive = P(C | +)
=
=
=0.3137
Probability that the male patient has prostate cancer if the PSA test comes back negative= P(C | -)
=
=
=0.2553
d)
The difference between P(C | +) and P(C | -) in parts (a) and (b) is significantly less than the difference between P(C | +) and P(C | -) in part (c)
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