An envelope is known to have come from either 'LONDON' or 'CLIFTON'. On the postal mark only two successive…
An envelope is known to have come from either 'LONDON' or 'CLIFTON'. On the postal mark only two successive letters ON are legible. The probability that the envelope comes from LONDON is
$\frac{12}{17}$
$\frac{5}{17}$
$\frac{3}{17}$
$\frac{2}{5}$
Solution
According to the given information, the number of two successive letters in word LONDON is 5, similarly in word CLIFTON is 6.
Let event $E_1$ is selecting word LONDON and event $E_2$ is selecting word CLIFTON.
So, $P\left(E_1\right)=\frac{1}{2}$ and $P\left(E_2\right)=\frac{1}{2}$
and the event $A$ is getting successive letter ON.
Then, $P\left(A / E_1\right)=\frac{2}{5}$ and $P\left(A / E_2\right)=\frac{1}{6}$.
$\therefore$ Required probability,
$
P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right) \cdot P\left(\frac{A}{E_1}\right)}{P\left(E_1\right) \cdot P\left(\frac{A}{E_1}\right)+P\left(E_2\right) \cdot P\left(\frac{A}{E_2}\right)}
$
$
\begin{aligned}
& =\frac{\frac{1}{2} \times \frac{2}{5}}{\left(\frac{1}{2} \times \frac{2}{5}\right)+\left(\frac{1}{2} \times \frac{1}{6}\right)}=\frac{\frac{2}{5}}{\frac{2}{5}+\frac{1}{6}} \\
& =\frac{2 \times 6}{12+5}=\frac{12}{17}
\end{aligned}
$
Hence, option (a) is correct