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

Let A₁ be the event that the Doctor diagnose the disease correctly and A₂ $={\overline{\mathrm{A}}}_1$
Let E be the event that a potent of the doctor having the particular disease survive by his treatment
$\begin{array}{r}\begin{array}{ll}\mathrm{P}\left({\mathrm{A}}_1\right)& =\frac{60}{100},\mathrm{P}\left({\mathrm{A}}_2\right)=\frac{40}{100},\mathrm{P}\left(\frac{\mathrm{E}}{{\mathrm{A}}_1}\right)\\ & =1-\mathrm{P}\left(\frac{\overline{\mathrm{E}}}{{\mathrm{A}}_1}\right)=1-\frac{40}{100}=\frac{60}{100}\end{array}\\ \text{ And }\mathrm{P}\left(\frac{\mathrm{E}}{{\mathrm{A}}_2}\right)=1-\mathrm{P}\left(\frac{\overline{\mathrm{E}}}{{\mathrm{A}}_2}\right)=1-\frac{70}{100}=\frac{30}{100}\\ \mathrm{P}\left(\mathrm{E}\right)=\mathrm{P}\left({\mathrm{A}}_1\right)\cdot \mathrm{P}\left(\frac{\mathrm{E}}{{\mathrm{A}}_1}\right)+\mathrm{P}\left({\mathrm{A}}_2\right)\cdot \mathrm{P}\left(\frac{\mathrm{E}}{{\mathrm{A}}_2}\right)\\ \mathrm{P}\left(\mathrm{E}\right)=\frac{60}{100}\left(\frac{60}{100}\right)+\frac{40}{100}\left(\frac{30}{100}\right)=\frac{48}{100}=\frac{12}{25}\\ \frac{2K}{25}=\frac{12}{25}\Rightarrow K=6\end{array}$