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. We are given the following probabilities:

1. \( P(M) = 0.6 \) (Probability that a married man watches the show)
2. \( P(W) = 0.5 \) (Probability that a married woman watches the show)
3. \( P(M|W) = 0.8 \) (Probability that a man watches the show given that his wife does watch)
4. \( P(W|M) = k \) (Probability that a wife watches the show given that her husband does watch)

We need to find \( \frac{1}{k} \).

### Step 1: Use Bayes' Theorem

According to Bayes' theorem, we have:

\[ P(W|M) = \frac{P(M|W) \cdot P(W)}{P(M)} \]

### Step 2: Substitute the known values

Substituting the known values into the equation:

\[ k = P(W|M) = \frac{P(M|W) \cdot P(W)}{P(M)} \]

Substituting the values we have:

\[ k = \frac{0.8 \cdot 0.5}{0.6} \]

### Step 3: Calculate \( k \)

Now, let's calculate \( k \):

\[ k = \frac{0.4}{0.6} = \frac{2}{3} \]

### Step 4: Find \( \frac{1}{k} \)

Now we need to find \( \frac{1}{k} \):

\[ \frac{1}{k} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

### Final Answer

Thus, the value of \( \frac{1}{k} \) is:

\[ \frac{1}{k} = \frac{3}{2} \]

---