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A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters $\mathrm{ON}$ are legible. The probability that it came from LONDON is
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The correct answer is:
$\frac{12}{17}$
We define the following events :
$A_1$ : Selecting a pair of consecutive letter from the word LONDON.
$A_2$ : Selecting a pair of consecutive letters from the word CLIFTON.
$\mathrm{E}$ : Selecting a pair of letters 'ON'.
Then $P\left(A_1 \cap E\right)=\frac{2}{5} ;$ as there are 5 pairs of consecutive letters out of which 2 are $\mathrm{ON}$.
$P\left(A_2 \cap E\right)=\frac{1}{6}$; as there are 6 pairs of consecutive letters of which one is ON.
$P\left(\frac{A_1}{E}\right)=\frac{P\left(A_1 \cap E\right)}{P\left(A_1 \cap E\right)+P\left(A_2 \cap E\right)}$
$A_1$ : Selecting a pair of consecutive letter from the word LONDON.
$A_2$ : Selecting a pair of consecutive letters from the word CLIFTON.
$\mathrm{E}$ : Selecting a pair of letters 'ON'.
Then $P\left(A_1 \cap E\right)=\frac{2}{5} ;$ as there are 5 pairs of consecutive letters out of which 2 are $\mathrm{ON}$.
$P\left(A_2 \cap E\right)=\frac{1}{6}$; as there are 6 pairs of consecutive letters of which one is ON.
$P\left(\frac{A_1}{E}\right)=\frac{P\left(A_1 \cap E\right)}{P\left(A_1 \cap E\right)+P\left(A_2 \cap E\right)}$
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