The probability that a married man watches a certain T.V. show is 0.6 and the probability that a married woman watches the show is 0.5. The probability that a man watches the show given that his wife does watch is 0.8. If the probability that a wife watches the show given that her husband does watch is k, then `(1)/(k)` is equal to

Answer

To solve the problem, we will use the concept of conditional probability. Let's denote the events as follows:

- Let \( A \) be the event that a married man watches the show.
- Let \( B \) be the event that a married woman watches the show.

Given data:
- \( P(A) = 0.6 \) (Probability that a married man watches the show)
- \( P(B) = 0.5 \) (Probability that a married woman watches the show)
- \( P(A | B) = 0.8 \) (Probability that a man watches the show given that his wife does watch)

We need to find \( \frac{1}{k} \) where \( k = P(B | A) \) (Probability that a wife watches the show given that her husband does watch).

### Step-by-Step Solution:

1. **Find \( P(A \cap B) \)**:
We can use the formula for conditional probability:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Rearranging gives:
\[ P(A \cap B) = P(A | B) \cdot P(B) \]
Substituting the known values:
\[ P(A \cap B) = 0.8 \cdot 0.5 = 0.4 \]

2. **Use the formula for \( P(B | A) \)**:
We can again use the conditional probability formula:
\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]
Substituting the values we have:
\[ P(B | A) = \frac{P(A \cap B)}{P(A)} = \frac{0.4}{0.6} \]

3. **Calculate \( P(B | A) \)**:
Simplifying the fraction:
\[ P(B | A) = \frac{0.4}{0.6} = \frac{4}{6} = \frac{2}{3} \]

4. **Find \( k \) and \( \frac{1}{k} \)**:
Since \( k = P(B | A) = \frac{2}{3} \), we need to find \( \frac{1}{k} \):
\[ \frac{1}{k} = \frac{1}{\frac{2}{3}} = \frac{3}{2} = 1.5 \]

### Final Answer:
\[ \frac{1}{k} = 1.5 \]