Answer:
a) Probability of prostrate cancer given a positive test is P(C|+) = 0.0213
b) Probability of cancer given a negative test is P(C|-) = 0.0161
c) Probability of prostrate cancer given a positive test is P(C|+) = 0.3137
d) Probability of cancer given a negative test is P(C|-) = 0.2553
Explanation:
Probability male patient has prostate cancer, P(C) = 0.02
Probability male patient does not have prostrate cancer P(C') = 1 - 0.02 = 0.98
Probability of a positive test given there is no cancer, i.e. P(false positive) = P(+|C') = 0.75
P(negative test given there is cancer) = P(false negative) = P(-|C) = 0.2
P(negative test given there is no cancer) is the complement of P(+|C') = P(-|C') = 1 - 0.75 = 0.25
Probability of positive test given there is prostrate cancer, P(+|C) is the complement of P(-|C), = 1 - 0.2 = 0.8.
a) Probability of prostrate cancer given a positive test is P(C|+)
According to Baye's theorem, P(C|+) = P(+|C)P(C)/P(+)
For P(+), we use the Law Of Total Probability: P(+) = P(+|C)P(C) + P(+|C')P(C')
P(+) = (0.8 * 0.02) + (0.75 * 0.98) = 0.751
Therefore, P(C|+) = P(+|C)P(C)/P(+)
P(C|+) = (0.8 * 0.02)/0.751 = 0.0213
b) Probability of cancer given a negative test is P(C|-)
According to Baye's theorem, P(C|-) = P(-|C)P(C)/P(-)
P(-) = P(-|C)P(C) + P(-|C')P(C')
P(-) = (0.2 * 0.02) + (0.25 * 0.98) = 0.249
Therefore, P(C|-) = (0.2 * 0.02)/0.249
P(C|-) = 0.0161
Part 2: Given the following;
Probability male patient has prostate cancer, P(C) = 0.3
Probability male patient does not have prostrate cancer P(C') = 1 - 0.3 = 0.70
Probability of a positive test given there is no cancer, i.e. P(false positive) = P(+|C') = = 0.75
P(negative test given there is cancer) = P(false negative) = P(-|C) = 0.2
P(negative test given there is no cancer) is the complement of P(+|C') = P(-|C') = 1 - 0.75 = 0.25
Probability of positive test given there is prostrate cancer, P(+|C) is the complement of P(-|C), = 1 - 0.2 = 0.8.
c) Probability of prostrate cancer given a positive test is P(C|+)
According to Baye's theorem, P(C|+) = P(+|C)P(C)/P(+)
For P(+), we use the Law Of Total Probability: P(+) = P(+|C)P(C) + P(+|C')P(C')
P(+) = (0.8 * 0.3) + (0.75 * 0.7) = 0.751
Therefore, P(C|+) = P(+|C)P(C)/P(+)
P(C|+) = (0.8 * 0.3)/0.765 = 0.3137
d) Probability of cancer given a negative test is P(C|-)
According to Baye's theorem, P(C|-) = P(-|C)P(C)/P(-)
P(-) = P(-|C)P(C) + P(-|C')P(C')
P(-) = (0.2 * 0.3) + (0.25 * 0.7) = 0.235
Therefore, P(C|-) = (0.2 * 0.3)/0.235
P(C|-) = 0.2553
