Question

Consider two events A and B, with $P\left(A\right)=.1$,$P\left(B\right)=.2$,and$P\left(A\cap B\right)=0$

a.Are A and B mutually exclusive?

b.Are A and B independent?

Short answer

1. The events are mutually exclusive.
2. The events are not independent.

Step-by-step solution

Mutually exclusive

The events are said to be mutually excusive that event that cannot occurs at the same time and having probability is zero.

The events are said independent if their probability does not affect each another. $P(A\cap B)=P\left(A\right)·P\left(B\right)$.

Find A and B mutually exclusive

Yes, event A and B are mutually exclusive because $P(A\cap B)=0$.

Hence, the events are mutually exclusive.

Showing A and B are not independent

b.

No, A, and B are not separate events.

Here,

$$\begin{gathered} P\left(A\right)·P\left(B\right)=0.1\times 0.2 \\ =0.02 \\ \ne 0 \end{gathered}$$

That is,$P\left(A\right)·P\left(B\right)\ne P\left(A\cap B\right)$

Therefore, the events are not independent.