A letter is known to have come from CHENNAI, JAIPUR, NAINITAL, DUBAI and MUMBAI. On the post mark only two consecutive letters AI are legible. Then, the probability that it is come from MUMBAI, is

Options

A. `(42)/(319)`
B. `(84)/(403)`
C. `(39)/(331)`
D. `(42)/(331)`

Answer

To find the probability that the letter came from Mumbai given that the visible letters are "AI", we need to follow these steps:

### Step 1: Identify the words and the pairs of letters
The words we have are:
- CHENNAI
- JAIPUR
- NAINITAL
- DUBAI
- MUMBAI

We need to identify how many pairs of consecutive letters can be formed from each word.

### Step 2: Count the pairs for each word
1. **CHENNAI**:
- Pairs: CH, HE, EN, NN, NA, AI (Total = 6 pairs)
2. **JAIPUR**:
- Pairs: JA, AI, IP, PU, UR (Total = 5 pairs)
3. **NAINITAL**:
- Pairs: NA, AI, IN, NI, IT, TA, AL (Total = 7 pairs)
4. **DUBAI**:
- Pairs: DU, UB, BA, AI (Total = 4 pairs)
5. **MUMBAI**:
- Pairs: MU, UM, MB, BA, AI (Total = 5 pairs)

### Step 3: Calculate the total number of occurrences of "AI"
Now, we need to find out how many times "AI" appears in the pairs:
- CHENNAI: 1 occurrence of "AI"
- JAIPUR: 1 occurrence of "AI"
- NAINITAL: 1 occurrence of "AI"
- DUBAI: 1 occurrence of "AI"
- MUMBAI: 1 occurrence of "AI"

### Step 4: Calculate the total number of pairs
Now, we sum the total pairs from all words:
- Total pairs from CHENNAI = 6
- Total pairs from JAIPUR = 5
- Total pairs from NAINITAL = 7
- Total pairs from DUBAI = 4
- Total pairs from MUMBAI = 5

Total pairs = 6 + 5 + 7 + 4 + 5 = 27 pairs

### Step 5: Calculate the probability of "AI" coming from Mumbai
The probability that the letter came from Mumbai given that "AI" is visible is calculated using the formula:

\[ P(Mumbai | AI) = \frac{P(AI | Mumbai) \cdot P(Mumbai)}{P(AI)} \]

Where:
- \(P(AI | Mumbai) = \frac{1}{5}\) (since there are 5 pairs in MUMBAI)
- \(P(Mumbai) = \frac{1}{5}\) (since there are 5 cities)
- \(P(AI) = \frac{1}{27}\) (total occurrences of "AI" in all pairs)

Substituting the values:

\[ P(Mumbai | AI) = \frac{\frac{1}{5} \cdot \frac{1}{5}}{\frac{1}{27}} = \frac{\frac{1}{25}}{\frac{1}{27}} = \frac{27}{25} \]

### Step 6: Final Calculation
Now we need to simplify this:

\[ P(Mumbai | AI) = \frac{27}{25} \cdot \frac{1}{27} = \frac{1}{25} \]

### Conclusion
Thus, the probability that the letter came from Mumbai given that "AI" is visible is:

\[ \frac{1}{25} \]