Probability: Mutually exclusive and independent events

​​In a nutshell

Events may be classed as mutually exclusive or independent. These have their own definitions and associated mathematical conditions.

Definitions

Two events are mutually exclusive if they cannot happen at the same time.

Two events are independent if one does not affect the other.

Mathematical conditions

​​​​Mutually exclusive events

If two events, $$A$$​ and $$B$$ cannot happen at the same time, the probability that they both occur is zero. Mathematically:

​$$P(A \cap B) =0$$​​

Also, the following diagram should convince you of the formula:

$$P(A \cup B) +P(A \cap B) = P(A)+P(B)$$​​

​

Putting these two formulae together, the two equivalent conditions for mutually exclusive events are:

​$$\boxed{\begin{aligned}P(A \cap B)&= 0\\P(A \cup B)&= P(A) +P(B)\end{aligned}}$$​​

Independent events

If two events, $$A$$ and $$B$$, are statistically independent, then:

​$$\boxed{P(A\cap B)=P(A)\times P(B)}$$​​

Example 1

Let $$A$$ and $$B$$ be two independent events. $$P(A)=0.4$$, $$P(B)=0.7$$. What is $$P((A \cap B)')$$?

Use the formula for independent events:

​$$\begin{aligned}P(A\cap B) &=P(A) \times P(B) \\&=0.4\times 0.7\\&=0.28\end{aligned}$$​​

Use the fact that the two events $$A \cap B$$ and $$(A \cap B)'$$ occupy the entire sample space:

​$$\begin{aligned}P(A \cap B) + P((A \cap B)') &= 1\\ P((A\cap B)') &=1-P(A \cap B)\\&=1-0.28\\&=0.72\end{aligned}$$​​

​$$P((A \cap B)')=\underline{0.72}$$​​

Example 2

Let $$A$$ and $$B$$ be two events. $$P(A)=0.35$$, $$P(B)=0.05$$, $$P(A \cup B) = 0.5$$. Are $$A$$ and $$B$$ mutually exclusive?

Use the condition for two events to be mutually exclusive:

​$$\begin{aligned}P(A)+P(B)&=P(A \cup B)\\0.35+0.05&=0.4\\&\neq 0.5\end{aligned}$$​​

The events are not mutually exclusive.