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Concept

Mutually Exclusive Events

Two events $A$ and $B$ are said to be mutually exclusive events or disjoint, if the events cannot occur simultaneously. This means that $A$ and $B$ have no common outcomes, implying that the probability of $A$ and $B$ is zero.

$P(A \text{ and } B)=0$

Thanks to the Addition Rule of Probability, it can be concluded that the probability of $A$ or $B$ is equal to the individual probabilities added together.

$P(A \text{ or } B)=P(A) + P(B)$

Below, some examples of mutually exclusive events are presented.

Three or more events are mutually exclusive if all pairs are mutually exclusive.

• The outcome of a dice roll is either $1,$ $2,$ $3,$ $4,$ $5,$ or $6.$
• In a soccer game, a team either wins or loses or the result is a draw.

Events can be mutually exclusive without being the only possible outcomes.

• A given number is either negative or positive.
• A random playing card drawn from a deck of cards is hearts or clubs.

Compare this concept with the concept of collectively exhaustive events, where the events cover all possible outcomes.

Events Are these Mutually Exclusive Events? Are these Collectively Exhaustive Events?
A given integer number is even or odd. Yes Yes
A given integer number is negative or positive. Yes No, it can also be zero.
A given integer number is greater than $3$ or less than $5$. No, $4$ is greater than $3$ and less than $5$. Yes
A given integer number is prime or even. No, $2$ is both prime and even. No, $9$ is neither prime nor even.