Rahul has to write a project, Probability that he will get a project copy is 'p', probability that he will get a blue pen is 'q' and probability that he will get a black pen is 1/2. If he can complete the project either with blue or with black pen or with both and probability that he completed the project is `1/2` then p(1+q) is

Options

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

Answer

To solve the problem step by step, we will analyze the situation regarding the probabilities of Rahul completing his project.

### Step 1: Understanding the Problem
Rahul needs a project copy and a pen (either blue or black) to complete his project. The probabilities given are:
- Probability of getting a project copy = \( p \)
- Probability of getting a blue pen = \( q \)
- Probability of getting a black pen = \( \frac{1}{2} \)

### Step 2: Defining the Events
To complete the project, Rahul can either:
1. Get a blue pen and not a black pen.
2. Get a black pen and not a blue pen.
3. Get both pens.

### Step 3: Writing the Probability of Completing the Project
The probability of completing the project can be expressed as:
\[ P(\text{Complete Project}) = P(\text{Copy}) \times \left( P(\text{Blue Pen}) \times P(\text{Not Black Pen}) + P(\text{Black Pen}) \times P(\text{Not Blue Pen}) + P(\text{Both Pens}) \right) \]

Where:
- \( P(\text{Not Black Pen}) = 1 - \frac{1}{2} = \frac{1}{2} \)
- \( P(\text{Not Blue Pen}) = 1 - q \)

### Step 4: Expanding the Probability Expression
Now, we can write the probability of completing the project as:
\[ P(\text{Complete Project}) = p \left( q \cdot \frac{1}{2} + \frac{1}{2} \cdot (1 - q) + pq \right) \]

### Step 5: Simplifying the Expression
Substituting the values:
\[ P(\text{Complete Project}) = p \left( \frac{q}{2} + \frac{1}{2} - \frac{q}{2} + pq \right) \]
This simplifies to:
\[ P(\text{Complete Project}) = p \left( \frac{1}{2} + pq \right) \]

### Step 6: Setting the Equation
We know that the probability of completing the project is given as \( \frac{1}{2} \):
\[ p \left( \frac{1}{2} + pq \right) = \frac{1}{2} \]

### Step 7: Solving for \( p(1 + q) \)
Dividing both sides by \( \frac{1}{2} \):
\[ p + p^2q = 1 \]
Rearranging gives:
\[ p(1 + q) = 1 \]

### Step 8: Conclusion
Thus, we find:
\[ p(1 + q) = 1 \]

### Final Answer
The value of \( p(1 + q) \) is \( 1 \).

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