Question

Suppose that the events A and B are disjoint and that

each has positive probability. Are A and B independent?

Short answer

If A and B are disjoint events with positive probabilities are never independent.

Step-by-step solution

Given information

Events A and B are disjoint.

Events A and B has positive probability.

Condition for independent events

Two events A and B would be independent if following condition holds true,

\(P\left( {A\;and\;B} \right) = P\left( A \right) \times P\left( B \right)\)

Check whether two disjoint events with positive probabilities are independent or not.

The disjoint events never occur at the same time.

If two events A and B are disjoint, then \(P\left( {A\;and\;B} \right) = 0\)

The probability of events A and B being positiveimplies that,

\(\begin{aligned}{l}P\left( A \right) > 0\\P\left( B \right) > 0\end{aligned}\)

It is known that,

\(\begin{aligned}{c}P\left( {A\;{\rm{and}}\;B} \right) = 0\\P\left( A \right) \times P\left( B \right) > 0 \Rightarrow P\left( A \right) \times P\left( B \right) \ne 0\end{aligned}\)

Hence, the two disjoint events A and B with positive probabilities are never independent.