NIMCET 2009 — Mathematics PYQ

Using the property of complementary events:

2. Find $P(A\cup B)$:

3. Find $P(B)$:

Using the addition theorem of probability:

Substitute the known values:

4. Check the properties:

• Equally Likely?

  $P(A)=\frac{3}{4}$ and $P(B)=\frac{1}{3}$. Since $P(A)\ne P(B)$, they are not equally likely.

• Mutually Exclusive?

  For events to be mutually exclusive, $P(A\cap B)$ must be $0$. Here, $P(A\cap B)=\frac{1}{4}$, so they are not mutually exclusive.

• Independent?

  For independence, $P(A\cap B)=P(A)\cdot P(B)$ must hold.

  $$P(A)\cdot P(B)=\frac{3}{4}\cdot \frac{1}{3}=\frac{1}{4}$$

  Since $P(A\cap B)=\frac{1}{4}$, the condition for independence is satisfied.

Final Answer:

The events $A$ and $B$ are independent but not equally likely. The correct option is (a).