Question

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

• A. $1/17$
• B. $12/17$
• C. $17/30$
• D. $3/5$

Answer 1

The correct option is B $12/17$

Let $E_1$ be letters coming from word LONDON

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

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

Let A be two consecutive letters on the envelope on ON.

If $E_1$ occurs then the letters ON come from the word LONDON. In this word there are 6 letters in which ON occurs twice. Considering one of the ON's as one object/letter there are now 5 letters.

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

Using Baye's theorem,

$P\left(\frac{E_1}{A}\right)=\frac{\frac{1}{2}\times \frac{2}{5}}{\frac{1}{2}\times \frac{2}{5}+\frac{1}{2}\times \frac{1}{6}}=\frac{1}{5}\times \frac{60}{12+5}=\frac{12}{17}$

is the required probability that it came from LONDON.