Question

Two fair dice are tossed. Let $A$ be the event that the first die shows an even number and $B$ be the event that the second die shows an odd number. The two events $A$ and $B$ are

• A. Mutually exclusive
• B. Independent and mutually exclusive
• C. Dependent
• D. None of the above

Answer 1

The correct option is D

None of the above

Find the probability:

Total number of sample space $=36$

Probability of event $A$ $=P\left(A\right)=\frac{18}{36}=\frac{1}{2}$

Probability of event $B$ $=P\left(B\right)=\frac{18}{36}=\frac{1}{2}$

Size of sample space of $P(A\cap B)$$=\left\{\right(2,3),(2,5),(2,1),(4,1),(4,3),(4,5),(6,1),(6,3),(6,5\left)\right\}=9$

The probability of an even number on the first and an odd number on the second is $=P(A\cap B)=\frac{9}{36}=\frac{1}{4}$

$\because P(A\cap B)=P\left(A\right)\times P\left(B\right)=\frac{1}{4}$

So, the given events $A$and $B$are independent events.

Option $A$: $A$and $B$ are not mutually exclusive, so this option is incorrect.

Option $B$: $A$and $B$ are independent but not mutually exclusive, so this option is incorrect

Option $C$: $A$and $B$ are not dependent events, so this option is incorrect.

Option $D$: since none of the given options is correct, so none of the above options is correct.

Hence, the correct option is (D).