A doctor diagnoses a certain illness correctly in 60% of the cases. A patient recovers with probability 0.9 if the diagnosis is correct and with probability 0.4 if it is not. A patient of this doctor did not recover. The probability that the diagnosis of this patient was correct is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{5}\)
Let \(E_1\) and \(E_2\) be the events that the diagnosis is correct and not correct, and let N be the event that the patient does not recover.
\(P(E_1) = 0.6,\quad P(N|E_1) = 1-0.9 = 0.1\)
\(P(E_2) = 0.4,\quad P(N|E_2) = 1-0.4 = 0.6\)
By the theorem of total probability:
\(P(N) = 0.6\times 0.1+0.4\times 0.6 = 0.06+0.24 = 0.30\)
By Bayes’ theorem:
\( \displaystyle P(E_1|N) = \frac{0.06}{0.30} = \frac{1}{5} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{4}{5}\) — Incorrect. This is \(\frac{0.24}{0.30}\), the chance that the diagnosis was not correct.
Option (2) \(\displaystyle\frac{3}{10}\) — Incorrect. This is P(N), the overall chance that a patient does not recover.
Option (3) \(\displaystyle\frac{3}{5}\) — Incorrect. This is the chance of a correct diagnosis before the result is known.