A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, and III are, respectively, p, q, and 1/2. then p(1+q)=

Options

A. `1//2`
B. 1
C. `3//2`
D. `3//4`

Explanation

To solve the problem, we need to find the value of \( p(1 + q) \) given the probabilities of passing tests I, II, and III as \( p \), \( q \), and \( \frac{1}{2} \) respectively.

### Step-by-Step Solution:

1. **Define Events**: Let \( A \), \( B \), and \( C \) represent the events of passing tests I, II, and III respectively. The probabilities are:
- \( P(A) = p \)
- \( P(B) = q \)
- \( P(C) = \frac{1}{2} \)

2. **Successful Conditions**: The student is successful if he passes either:
- Tests I and II (events \( A \) and \( B \))
- Tests I and III (events \( A \) and \( C \))

Therefore, the successful event can be represented as:
\[ P(\text{Success}) = P(A \cap B) + P(A \cap C) \]

3. **Calculate Probabilities**:
- The probability of passing both tests I and II:
\[ P(A \cap B) = P(A) \cdot P(B) = p \cdot q \]
- The probability of passing both tests I and III:
\[ P(A \cap C) = P(A) \cdot P(C) = p \cdot \frac{1}{2} \]

4. **Combine Probabilities**:
\[ P(\text{Success}) = P(A \cap B) + P(A \cap C) = pq + p \cdot \frac{1}{2} \]

5. **Set Up the Equation**: Since the student is successful if he passes either of the conditions, we can express this as:
\[ P(\text{Success}) = pq + \frac{p}{2} \]

6. **Total Probability of Success**: The total probability of success must equal the probability of passing at least one of the tests. The probability of passing at least one of the tests can be calculated as:
\[ P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) \]
However, since we are only interested in the successful conditions defined earlier, we can simplify our approach.

7. **Equating to 1**: From the problem, we can set the successful probability equal to 1:
\[ pq + \frac{p}{2} = 1 \]

8. **Rearranging the Equation**:
\[ pq + \frac{p}{2} = 1 \implies p(1 + q) = 1 \]

9. **Final Result**: Thus, we find:
\[ p(1 + q) = 1 \]

### Conclusion:
The value of \( p(1 + q) \) is \( 1 \).