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Probability: Mutually exclusive and independent events

Probability: Mutually exclusive and independent events

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Explainer Video

Tutor: Bilal

Summary

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, AAA​ and BBB cannot happen at the same time, the probability that they both occur is zero. Mathematically:

​P(A∩B)=0P(A \cap B) =0P(A∩B)=0​​


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


P(A∪B)+P(A∩B)=P(A)+P(B)P(A \cup B) +P(A \cap B) = P(A)+P(B)P(A∪B)+P(A∩B)=P(A)+P(B)​​
Maths; Probability; KS5 Year 12; Probability: Mutually exclusive and independent events

​

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


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


Independent events

If two events, AAA and BBB, are statistically independent, then:


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


Example 1

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


Use the formula for independent events:

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


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

​P(A∩B)+P((A∩B)′)=1P((A∩B)′)=1−P(A∩B)=1−0.28=0.72\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∩B)+P((A∩B)′)P((A∩B)′)​=1=1−P(A∩B)=1−0.28=0.72​​​


​P((A∩B)′)=0.72‾P((A \cap B)')=\underline{0.72}P((A∩B)′)=0.72​​​


Example 2

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


Use the condition for two events to be mutually exclusive:

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


The events are not mutually exclusive.



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Calculating probability

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The AND / OR rules

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FAQs - Frequently Asked Questions

What is the condition for independence?

Two events, A and B, are independent if P(A n B) = P(A)xP(B).

What does it mean for two events to be independent?

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

What does it means for two events to be mutually exclusive?

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

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