Question

For the following question, enter the correct numerical value upto TWO decimal places. If the numerical value has more than two decimal places, round-off the value to TWO decimal places. (For example: Numeric value 5 will be written as 5.00 and 2.346 will be written as 2.35) An envelope is known to have come from either `LONDON` OR `CLIFTON`. On the postal card only two successive letters `ON` are visible. The probability that the envelope comes from LONDON is \( \dfrac{12}{__} \).

Hint:

When partial information is given, use conditional probability or Bayes’ theorem by comparing the number of favorable outcomes to the total possible outcomes.

Answer 1

Correct Answer: 17

Step 1: Write the word LONDON: \[ L\ O\ N\ D\ O\ N \] The total number of successive letter pairs is: \[ 5 \] The pair ON appears twice. \[ P(\text{ON} \mid \text{LONDON}) = \frac{2}{5} \]
Step 2: Write the word CLIFTON: \[ C\ L\ I\ F\ T\ O\ N \] The total number of successive letter pairs is: \[ 6 \] The pair ON appears once. \[ P(\text{ON} \mid \text{CLIFTON}) = \frac{1}{6} \]
Step 3: Assuming the envelope is equally likely to come from either place: \[ P(\text{LONDON}) = P(\text{CLIFTON}) = \frac{1}{2} \]
Step 4: Using Bayes’ theorem: \[ P(\text{LONDON} \mid \text{ON}) = \frac{\frac{2}{5}}{\frac{2}{5} + \frac{1}{6}} \]
Step 5: Simplify: \[ \frac{2}{5} = \frac{12}{30}, \quad \frac{1}{6} = \frac{5}{30} \] \[ P(\text{LONDON} \mid \text{ON}) = \frac{12}{12+5} = \frac{12}{17} \] Final Answer (up to two decimal places): \[ \boxed{17.00} \]