Question

A letter is known to have come from either LONDON (or) CLIFTON. On the envelope just two consecutive letters 'ON' are visible. The probability that the letter comes from LONDON is

• A. $\frac{2}{5}$
• B. $\frac{3}{17}$
• C. $\frac{5}{17}$
• D. $\frac{12}{17}$

Answer 1

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

Let $E_1\to$ Letters came from LONDON.

$E_2\to$ Letters came from CLIFTON.

Since, the letters have to come from either of these two words.

$\Rightarrow$ $P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2}$

Let $A\to$ Two consecutive letters on the envelope are $ON$

If $E_1$ occurs then the letters $ON$ come from LONDON.

Considering one of $O{N}^{\prime }s$ as one object.

$\Rightarrow$ $P\left(\frac{A}{E_1}\right)=\frac{2}{5}$

If $E_2$ occurs from the word CLIFTON.

$\Rightarrow$ $P\left(\frac{A}{E_2}\right)=\frac{1}{6}$

Using Bayes theorem.

$\Rightarrow$ $P\left(\frac{E_1}{A}\right)=\frac{(\frac{1}{2}\times \frac{2}{5})}{(\frac{1}{2}\times \frac{2}{5}+\frac{1}{2}\times \frac{1}{6})}$

$=\frac{2}{5}\times \frac{30}{17}$

$=\frac{12}{17}$