Question

A letter is known to have come either from $LONDON$ or $CLIFTON$; on the postmark only the two consecutive letters $ON$ are ellegible. The probability that it came from $LONDON$ is

• A. $\frac{5}{17}$
• B. $\frac{12}{17}$
• C. $\frac{17}{30}$
• D. $\frac{3}{5}$

Answer 1

The correct option is B $\frac{12}{17}$

We define the following events:
$A_1:$ Selecting a pair of consecutive letters from the word $LONDON$
$A_2:$ Selecting a pair of consecutive letters from the word $CLIFTON$
$E:$ Selecting a pair of letters $ON$
Then $P(A_1\cap E)=\frac{2}{5}$, as there are $5$ pairs of consecutive letters out of which $2$ are $ON$.
$P(A_2\cap E)=\frac{1}{6}$, as there are $6$ pairs of consecutive letters of which $1$ is $ON$.
So, required probability $=P\left(\frac{A_1}{E}\right)$
$\Rightarrow P\left(\frac{A_1}{E}\right)=\frac{P(A_1\cap E)}{P(A_1\cap E)+P(A_1\cap E)}=\frac{\frac{2}{5}}{\frac{2}{5}+\frac{1}{6}}=\frac{12}{17}$