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 legible letters are "AI", we can follow these steps:

### Step 1: Identify the cities and their letters
The cities from which the letter could have come are:
- Chennai
- Jaipur
- Nainital
- Dubai
- Mumbai

### Step 2: Identify occurrences of "AI"
Next, we need to determine which of these cities contain the consecutive letters "AI":
- **Chennai**: Contains "AI" (in the sequence "NAI")
- **Jaipur**: Does not contain "AI"
- **Nainital**: Contains "AI" (in the sequence "NAI")
- **Dubai**: Contains "AI" (in the sequence "BAI")
- **Mumbai**: Contains "AI" (in the sequence "MBAI")

Thus, the cities that contain "AI" are Chennai, Nainital, Dubai, and Mumbai.

### Step 3: Count the total occurrences of "AI"
Now, let's count the total occurrences of "AI" in the cities:
- Chennai: 1 occurrence
- Nainital: 1 occurrence
- Dubai: 1 occurrence
- Mumbai: 1 occurrence

Total occurrences of "AI" = 4.

### Step 4: Count the favorable outcomes for Mumbai
The favorable outcome for the letter coming from Mumbai is 1 occurrence of "AI".

### Step 5: Calculate the probability
The probability that the letter came from Mumbai given that "AI" is legible can be calculated using the formula:

\[ P(\text{Mumbai} | \text{AI}) = \frac{\text{Number of favorable outcomes for Mumbai}}{\text{Total occurrences of AI}} \]

Substituting the values we found:

\[ P(\text{Mumbai} | \text{AI}) = \frac{1}{4} \]

### Final Answer
Thus, the probability that the letter came from Mumbai is:

\[ \frac{1}{4} \]

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