Mutually Exclusive Events

Mutually exclusive events are those events that cannot occur at the same time. For example, when a coin is tossed, the result will be either heads or tails. It is impossible to get both results. Such events are also called disjoint events because they cannot happen simultaneously. If A and B are mutually exclusive events, the probability is given as $P (A \cap B) = 0$ .

How to find mutually exclusive events

In probability, the specific addition rule is valid when two events are mutually exclusive. It states that the probability of either event occurring is the sum of the probabilities of each event occurring. If A and B are said to be mutually exclusive events, the probability of an event A occurring or the probability of an event B occurring that is $P (A \cap B)$ formula is given as $P (A) + P (B)$ .

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

If the events A and B are not mutually exclusive, the probability of getting event A or B that is $P (A \cup B)$ formula is given as follows:

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

In this formula, $P (A \cap B)$ is the probability of both A and B occurring.

Real-life examples of mutually exclusive events

When tossing a coin, the possibility of getting heads and tails are mutually exclusive events because the probability of getting both heads and tails simultaneously is 0.
With a six-sided die, the events "2" and "5" are mutually exclusive. You cannot roll both 2 and 5 on a toss of one die.
In a deck of 52 cards where we draw a single card, drawing a red card and drawing a club are mutually exclusive because all clubs are black.

Dependent and independent events

Two events are dependent if the occurrence of one changes the probability of the other. Two events are independent if the probability of one does not affect the probability of the other. If two events are mutually exclusive, they are not independent. Similarly, independent events cannot be mutually exclusive. Unless one of the probabilities is zero.

Rules for mutually exclusive events

In probability theory, two events are mutually exclusive or disjoint if they cannot occur at the same time. A clear case is the result of a single coin toss. The toss can end up as heads or tails, but not both.

While tossing the coin, both outcomes (heads and tails) are collectively exhaustive. This means at least one of the results must happen, so these two possibilities collectively exhaust all of the possibilities.

Not all mutually exclusive events are collectively exhaustive. For example, when you toss a die, the results 2 and 4 are mutually exclusive because you cannot get both at the same time. However, they are not collectively exhaustive, because there is also the possibility that you could roll a 1, 3, 5, or 6.

From the definition of mutually exclusive events, the following rules for probability are concluded:

$P (A \cap B) = 1$
$P (A \cup B) = 0$

Practice solving a probability problem involving mutually exclusive events

From a group of 6 freshmen and 5 sophomores, 3 students are to be selected at random to form a committee. What is the probability that at least two freshmen will be selected?

The committee will have at least two freshmen if either 2 freshmen and 1 sophomore are selected (event A) or 3 freshmen are selected (event B). Since event A and event B are mutually exclusive:

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

$P (A) = \frac{{}_{6}C_{2} \cdot {}_{5}C_{1}}{{}_{11}C_{3}} = \frac{15 \cdot 5}{165} = \frac{75}{165} = \frac{5}{11}$

$P (B) = \frac{{}_{6}C_{3}}{{}_{11}C_{3}} = \frac{20}{165} = \frac{4}{33}$

$P (A \cup B) = \frac{\frac{5}{11}}{+} = \frac{\frac{15}{33}}{+} = \frac{19}{33}$

So the probability that the committee will have at least two freshmen is $\frac{19}{33}$ .

Topics related to the Mutually Exclusive Events

Independent Events

Complement of an Event

Addition Rule of Probability

Flashcards covering the Mutually Exclusive Events

Statistics Flashcards

Common Core: High School - Statistics and Probability Flashcards

Practice tests covering the Mutually Exclusive Events

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

Get help learning about mutually exclusive events

The probability and statistics subject of mutually exclusive events can be tricky, especially once you go beyond collectively exhaustive outcomes. If you find yourself having trouble solving some of the more complex equations that come up while studying mutually exclusive events, a good idea is to work with an expert tutor who knows the subject well. A private tutor can work at your pace, taking the extra time to work on the concepts or equations that are especially challenging for you and skimming quickly through those concepts that you learn easily and quickly. They can work with you to discover your learning style and present lessons using the appropriate teaching method so you have an easier time understanding the information they are conveying.

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