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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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.

- A given integer number is either even or odd.
- The outcome of a coin toss is either heads or tails.

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. |