One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is :

Options

A. `1//2`
B. `1//3`
C. `2//5`
D. `1//5`

Answer

To solve the problem step by step, we need to find the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife.

### Step 1: Define the Events
Let:
- Event A: The Indian man is seated adjacent to his wife.
- Event E: Each American man is seated adjacent to his wife.

### Step 2: Calculate P(A ∩ E)
We first need to calculate the probability of both events A and E occurring together, denoted as P(A ∩ E).

1. **Couples as Entities**: Treat each couple as a single entity. We have 4 American couples and 1 Indian couple, making a total of 5 entities.
2. **Arranging Couples**: The number of ways to arrange 5 couples around a circular table is given by (5 - 1)! = 4!.
3. **Arranging Individuals within Couples**: Each couple can switch places, so for 5 couples, the number of arrangements is \(2^5\).

Thus, the total arrangements for P(A ∩ E) is:
\[ P(A ∩ E) = 4! \times 2^5 \]

### Step 3: Calculate P(E)
Now we need to calculate the probability of event E occurring alone.

1. **American Couples as Entities**: Treat each American couple as a single entity, and the Indian man and his wife as separate entities. This gives us 4 American entities and 2 Indian entities, making a total of 6 entities.
2. **Arranging Entities**: The number of ways to arrange 6 entities around a circular table is (6 - 1)! = 5!.
3. **Arranging Individuals within American Couples**: Each of the 4 American couples can switch places, which gives us \(2^4\) arrangements.

Thus, the total arrangements for P(E) is:
\[ P(E) = 5! \times 2^4 \]

### Step 4: Calculate the Conditional Probability P(A | E)
Using the formula for conditional probability:
\[ P(A | E) = \frac{P(A ∩ E)}{P(E)} \]

Substituting the values we calculated:
\[ P(A | E) = \frac{4! \times 2^5}{5! \times 2^4} \]

### Step 5: Simplify the Expression
Now, we simplify the expression:
1. \(4! = 24\)
2. \(5! = 120\)
3. \(2^5 = 32\)
4. \(2^4 = 16\)

Thus:
\[ P(A | E) = \frac{24 \times 32}{120 \times 16} \]

Calculating this gives:
\[ = \frac{768}{1920} = \frac{2}{5} \]

### Conclusion
The conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is:
\[ \frac{2}{5} \]