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 $$\frac{1}{2}$$. If he can complete the project either with blue or with black pen or with both and probability that he completed the project is $$\frac{1}{2}$$ then $$p\left( {1 + q} \right)$$   is :

Solution :
Lets define the events as
Probability of getting project copy $$\left( A \right) = p$$
Probability of getting blue pen $$\left( B \right) = q$$
Probability of getting black pen $$\left( C \right) = \frac{1}{2}$$
Then,
$$\eqalign{ & {\text{ }}p\left( {AB\overline C } \right) + p\left( {AC\overline B } \right) + p\left( {ABC} \right) = \frac{1}{2} \cr & p.q.\frac{1}{2} + p.\frac{1}{2}\left( {1 - q} \right) + p.q.\frac{1}{2} = \frac{1}{2} \cr & \therefore \,pq + p - pq + pq = 1 \cr & \therefore \,p\left( {1 + q} \right) = 1 \cr} $$