Question

The chance that a doctor will diagnose a certain disease correctly is $60\%$. The chance that a patient of a doctor will die by this treatment after correct diagnosis is $40\%$ and the chances of death by wrong diagnosis is $70\%$. The chances that the patient of a doctor having the particular disease will survive is $\frac{2K}{25}$. Then $K=$

Answer 1

Let $E_1$ be the event that the doctor diagnose the disease correctly and $E_2$ Be the event that the doctor diagnose the disease incorrectly.
Let $E$ be the event that a patient of the doctor having the particular disease survive by his treatment.
$P\left(E_1\right)=\frac{60}{100},P\left(E_2\right)=\frac{40}{100},P\left(E/E_1\right)=1-\frac{40}{100}=\frac{60}{100}$
$P\left(E/E_2\right)=1-\frac{70}{100}=\frac{30}{100}$
$P\left(E\right)=P\left(E_1\right).P\left(E/E_1\right)+P\left(E_2\right).P\left(E/E_2\right)$
$=(\frac{60}{100}\times \frac{60}{100}+\frac{40}{100}\times \frac{30}{100})=\frac{36+12}{100}=\frac{12}{25}=\frac{2K}{25}$
$\therefore K=6$