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Example A patient has complained of certain symptoms. From the experience of doctors, we know that about 5% of people in the same age group have cancer when they complain of the condition. On the other hand, a detailed examination shows a positive reaction of 85% for true cancer patients and a positive reaction of 5 % for non-cancer patients. If a patient gives a positive result on the work-up, find out the probability that the patient has cancer. Answer Let A = 「true cancer patient」， B = 「tested positive by detailed examination. If the patient shows a positive reaction as a result of the detailed examination, the probability of being a cancer patient is calculated. The fact that a patient who shows a positive reaction as a result of a detailed examination is a cancer patient is to find P(cancer patient | positive by detailed examination)．

 positive negative
cancer patient  0.85  0.15
non cancer patient 0.05  0.95

Here， P(positive by detailed exam | cancer patient) = 0.85, P(positive by detailed exam | non cancer patient) = 0.05, P(cancer patient) = 0.05

P (A|B) is a conditional probability．

P (A|B) = \frac{P (B ∩ A)}{P (B)}

Note that

B = B ∩ Ω = B ∩ (A ∪ Ā) = (B ∩ A) ∪ (B ∩ Ā)

By the problem， P (B ∩ A) = P (A ∩ B) = P (A)P (B|A) = 0.05 ∗ 0.85, P (B ∩ Ā) = P (Ā ∩ B) = P (Ā)P (B|Ā) = 0.95 ∗ 0.05. Therefore，

P (A|B) = \frac{P (B ∩ A)}{P (B)}
= \frac{P (B ∩ A)}{P (B ∩ A) + P (B ∩ Ā)}
= \frac{0.05 ∗ 0.85}{0.05 ∗ 0.85 + 0.95 ∗ 0.05} = \frac{0.425}{0.9}