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The chance that doctor A will diagnose the disease X correctly is 60%. The chance that a patient will die by his treatment after correct diagnosis is 40% and the chance of death after wrong diagnosis is 70%. A patient of doctor A who had disease X died. The probability that his disease was diagnosed correctly is

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$\frac{5}{13}$

$\frac{6}{13}$

$\frac{1}{13}$

$\frac{2}{13}$

answer is B.

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Detailed Solution

A – diagnose correctly $\Rightarrow P (A) = \frac{60}{100}$

$P (\bar{A}) = \frac{40}{100}$

$B_{1}$ - death after correct diagnosis $\Rightarrow P (B_{1}) = \frac{40}{100}$

$B_{2}$ - death after wrong diagnosis $\Rightarrow P (B_{2}) = \frac{70}{100}$

$P (\frac{B_{1}}{A}) = \frac{P (A) . P (\frac{B_{1}}{A})}{P (A) . P (\frac{B_{1}}{A}) + P (\bar{A}) . P (\frac{B_{2}}{A})}$

$= \frac{\frac{60}{100} \times \frac{40}{100}}{\frac{60}{100} . \frac{40}{100} + \frac{40}{100} \times \frac{70}{100}} = \frac{4 \times 6}{24 + 28} = \frac{6}{6 + 7} = \frac{6}{13}$

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