Question

The chance that Doctor $A$ will diagnose disease $X$ correctly is 60%. The chance that a patient will die by his treatment after the correct diagnosis is 40% and the chance of death after the wrong diagnosis is 70%. A patient of Doctor $A$ who had disease $X$ died. The probability that his disease was diagnosed correctly is
A. $\dfrac{5}{{13}}$
B. $\dfrac{6}{{13}}$
C. $\dfrac{2}{{13}}$
D. $\dfrac{7}{{13}}$

Answer 1

In order to find the required probability first we will be given events and conditions as a variable and we will proceed further by using Bayes’ theorem which describes the probability of an event based on prior knowledge of conditions that might be related to the event.

Complete step-by-step solution:
Let us define the following events as:
${E_1}$: Disease is diagnosed correctly by doctor $A$.
${E_2}$: Disease is not diagnosed correctly by doctor $A$.
$B$: A patient (of doctor $A$) who has disease $X$ dies.
Given that, $P\left( {{E_1}} \right) = \dfrac{{60}}{{100}} = 0.6$
We know that, $P\left( {{E_1}} \right) + P\left( {{E_2}} \right) = 1$
So, we have $P\left( {{E_2}} \right) = 1 - P\left( {{E_1}} \right) = 1 - 0.6 = 0.4$
Also given that, the probability of $B$given when ${E_1}$ is true is $P\left( {\dfrac{B}{{{E_1}}}} \right) = \dfrac{{40}}{{100}} = 0.4$
Probability of $B$given when ${E_2}$ is true is $P\left( {\dfrac{B}{{{E_2}}}} \right) = \dfrac{{70}}{{100}} = 0.7$
Now, we have to find the probability of ${E_1}$ when $B$ is true i.e., $P\left( {\dfrac{{{E_1}}}{B}} \right)$
By Bayes’ Theorem, we have

$$\Rightarrow P\left( {\dfrac{{{E_1}}}{B}} \right) = \dfrac{{P\left( {{E_1}} \right)P\left( {\dfrac{B}{{{E_1}}}} \right)}}{{\,P\left( {{E_1}} \right)P\left( {\dfrac{B}{{{E_1}}}} \right) + P\left( {{E_2}} \right)P\left( {\dfrac{B}{{{E_2}}}} \right)}} \\\ \Rightarrow P\left( {\dfrac{{{E_1}}}{B}} \right) = \dfrac{{0.6 \times 0.4}}{{0.6 \times 0.4 + 0.4 \times 0.7}} \\\ \Rightarrow P\left( {\dfrac{{{E_1}}}{B}} \right) = \dfrac{{0.24}}{{0.24 + 0.28}} \\\ \therefore P\left( {\dfrac{{{E_1}}}{B}} \right) = \dfrac{{0.24}}{{0.52}} = \dfrac{6}{{13}} \\\$$

Therefore, the correct option is B.

Note: Bayes' theorem is a mathematical formula for determining conditional probability. Conditional probability is the likelihood of an outcome occurring, based on a previous outcome occurring. Bayes' theorem provides a way to revise existing predictions or theories (update probabilities) given new or additional evidence.