NIMCET 2008 — Mathematics PYQ

First, we find the probability of event $A$ using the complement rule:

• $P(A)=1-P(A^{\prime })=1-0.3=0.7$

• $P(B)=0.5$

• $P(B^{\prime })=1-P(B)=1-0.5=0.5$

• $P(A\cap B)=0.3$

2. Apply the Conditional Probability Formula

The formula for conditional probability $P(X|Y)$ is $\frac{P(X\cap Y)}{P(Y)}$. Here, our $X=A$ and $Y=A\cup B^{\prime }$.

3. Simplify the Numerator and Denominator

The Numerator: Using the absorption law in set theory, $A\cap (A\cup B^{\prime })=A$. Therefore:

The Denominator:

Using the addition theorem for $P(A\cup B^{\prime })$:

We know $P(A\cap B^{\prime })=P(A)-P(A\cap B)$, so:

Now substitute back into the denominator:

4. Final Calculation

Conclusion

Since $\frac{7}{8}$ is not among the options (a), (b), or (c), the correct choice is (d) None.