An unbiased die is thrown twice. Let the event $A$ be ‘odd number on the first throw’ and $B$ the event ‘odd number on the second throw’. Check the independence of the events $A$ and $B$.

Yes, they are independent.

The correct answer is Option (1) → Yes, they are independent. ##

If all the 36 elementary events of the experiment are considered to be equally likely, we have

$P(A) = \frac{18}{36} = \frac{1}{2} \text{ and } P(B) = \frac{18}{36} = \frac{1}{2}$

Also $P(A \cap B) = P(\text{odd number on both throws})$

$= \frac{9}{36} = \frac{1}{4}$

Now $P(A) \cdot P(B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Clearly $P(A \cap B) = P(A) \times P(B)$

Thus, $A$ and $B$ are independent events.