Mathematics - Probability Question with Solution | TestHub

Let $E_1$ denote the event "a coin with head on both sides is selected" and ${\mathrm{E}}_2$ denotes the event " a fair coin is selected". Let $A$ be the event " he toss, results in heads". $$\begin{array}{l}\therefore \mathrm{P}\left({\mathrm{E}}_1\right)=\frac{1}{n+1},\mathrm{P}\left({\mathrm{E}}_2\right)=\frac{n}{n+1}\text{ and }\\ \mathrm{P}\left(\frac{\mathrm{A}}{{\mathrm{E}}_1}\right)=1,\mathrm{P}\left(\frac{\mathrm{A}}{{\mathrm{E}}_2}\right)=\frac{1}{2}\\ \therefore \mathrm{P}(\mathrm{A})=\mathrm{P}\left({\mathrm{E}}_1\right)\mathrm{P}\left(\frac{\mathrm{A}}{{\mathrm{E}}_1}\right)+\mathrm{P}\left({\mathrm{E}}_2\right)\mathrm{P}\left(\frac{\mathrm{A}}{{\mathrm{E}}_2}\right)\\ \Rightarrow \frac{7}{12}=\frac{1}{n+1}\times 1+\frac{n}{n+1}\times \frac{1}{2}\\ \Rightarrow 14n+14=24+12n\Rightarrow n=5\end{array}$$