Mutually exclusive events: definition, rules and formula (2023)

Definition of mutually exclusive events

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Any two events that cannot occur simultaneously are referred to as mutually exclusive events with respect to each other. Such events are also nameddisjoint events. Events that are mutually exclusive always have different outcomes. Mathematically, mutually exclusive events where the probability of the events in question is zero or has no value.

How do you find and show that two events are mutually exclusive?

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If in the mathematical probability theorem 'A'e'B'are two mutually exclusive events, so their probability can be written asP(AB) YouP(AYou b).This symbol ∩ means 'e' and the probability of occurrence of event 'W' and event 'X' would be zero. For this reason,

P(Wx) = 0

The probability of two events "A" and "B" is calculated as follows:

P(AB) = P(A) + P(B)

If the result of P(A) + P(B) = 0, then these two events are mutually exclusive.

For events that are not mutually exclusive, the probability P(A ∪ B) is calculated using the following formula:

P (A ∪ B) = P(A) + P(B) – P (A e B)

For more information, see:

Relevant concepts
Combinatorics Probability multiplication rule define operations

Mutually exclusive eventsRules and formula

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As mentioned earlier, mutually exclusive events do not occur simultaneously. This does not mean that events that are mutually exclusive are altogether exhaustive.

For example, a traffic light that only shows green and red light. In both cases the signal light is either green or red, but not both. These are events that are mutually exclusive. Furthermore, there is no possibility of a third outcome. So these are exhausting events overall. Collectively exhaustive possibilities are those where there is no chance of a different outcome or probability.

Mutually exclusive events: definition, rules and formula (1)

Street sign showing red and green light

For example,Imagine throwing a Ludo die, the events of getting the numbers 1 and 3 are mutually exclusive as both cannot happen together. But these two possibilities taken together are not exhaustive as there is a good chance of getting a third possible result (2, 4, 5 or 6).

From the above definition of mutually exclusive events, one can deduce certain rules for any two probabilities related to it.

Addition rule - P(X + Y) = 1

Subtraction rule - P( X U Y) = 0

Multiplication Rule - P( X ∩ Y) = 0

With certain changes in state, there are a variety of events.

For example, a coin that has heads on both sides of the coin or tails on both sides. No matter how many times you toss it, heads (for the first coin) and tails (for the second coin) will always appear.

When examining the sample space of such an experiment, it will always be heads for the first coin and tails for the second. Such events have unique points in sample space and are invoked"Simple Events". These two example event types are always mutually exclusive.

Mutually exclusive events: Formula

These mutually exclusive event formulas can be used to solve questions based on the probability of mutually exclusive events.

The probability of two events stating that A and B are mutually exclusive is represented as

(Video) Mutually Exclusive vs. Independent Events EXPLAINED in 4 minutes

  • A and B

The intersection set between A and B is equal to {null}. So P(A and B) = 0.

Because if two events cannot happen simultaneously, there will obviously be nothing in common.

  • a or B

P(A or B) = P(A) + P(B) - P(A and B)

The probability of the union of two mutually exclusive events is derived by adding the probabilities of the events separately.

Dependent and independent events

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Two events are classified asdependent eventswhen one event affects the probability of another event occurring. On the other hand, two events that do not affect the probability of the other in any way are calledstandalone events.Independent events are mutually exclusiveifthey affect the likelihood of each other. Also, independent events can never be mutually exclusive.

For more information, see: events in probability

Examples of mutually exclusive events

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  • One person can turn left or right at the same time. Therefore it is an example of mutually exclusive events.
  • In a deck of cards, kings and aces are mutually exclusive since you can draw an ace card or a king card at a time.
  • The occurrence of day and night is another excellent example of how nature is mutually exclusive.
  • A gas stove cannot be turned on and off at the same time, so these events are mutually exclusive.

Conditional probability for mutually exclusive events

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Another special case of probability isconditional probabilitywhere if one event occurs, only the other event occurs or not. Thus, the second event is based on the occurrence condition of the first. Such events are calledconditional events.

The conditional probability for two independent events "A" and "B" where event B resulted in event A is denoted by the expression P(B|A) and defined by the following equation:

P(B|A)= P(A ∩ B)/P(A)

The above equation using the rule of multiplication becomes: P(A ∩ B) = 0

P(B|A)= 0/P(A)

Therefore, the conditional probability for mutually exclusive events is:

P (B | A) = 0

things to remember

  • Mutually exclusive events are those events that cannot occur simultaneously with respect to each other. They are also known as disjoint events.
  • Mutually exclusive events are dependent events because they affect each other.
  • We can show that given two events A and B, they are mutually exclusive by proving their probabilityP(AB) YouP(AOr B) is zero.
  • The probability of the union of two mutually exclusive events is derived by adding the probabilities of the two events separately.

sample questions

From the given set of events, indicate which are mutually exclusive: (4 points)
(i) Drawing a king from a deck and a heart card.
(i) Rolling 1 and 3 in a dice roll.
(iii) Getting tails from a coin with tails on both sides.
(iv) Rolling 3 and 4 when rolling the dice.


  1. It's not an example of a mutually exclusive event, because in a deck there is a "King of Hearts" card that contains both a king and hearts, so it's possible to get a king and hearts card at the same time . Therefore, it is not a mutually exclusive event.
  2. This is an example of a mutually exclusive event. When a die is rolled, you cannot roll a 1 and a 3 at the same time.
  3. It's not a mutually exclusive event as the coin has tails on both sides and only the event is mentioned in the question. Without a second relatable event, there is no way to decide whether or not it is a mutually exclusive event. It's a simple event.
  4. This is a mutually exclusive event. There is no way to get 1 and 3 at the same time here.

Task Three coins are tossed at the same time. We say A as the event of getting at least 2 heads. Likewise, B denotes the event of getting a coin and C the event of getting heads on the second coin. Which of these are mutually exclusive? (5 points)

Or.First, create a sample room for each event. For event 'A' we need to get at least two heads. Therefore, include all events that have two or more heads.

Or it can be written as:


This set A contains 4 elements or events, so n(A) = 4

Similarly for Event B it can be written as:

(Video) Multiplication & Addition Rule - Probability - Mutually Exclusive & Independent Events

B = {TTT} e n(B) = 1

This set has only one element.

In the same way,

C = {THT, HHH, HHT, THH} und n(C) = 4

Hence B & C and A & B are mutually exclusive as they have nothing at their intersection.

Question The odds of the 3 teams a, b, c to win a soccer game are 1/3, 1/5 and 1/9 respectively. Find the probability of that (3 points)
a] of the three teams, team a or team b wins
b] Team a or team b or team c wins
c] Neither team will win the game
d] Neither Team A nor Team B will win the game

Or.a) P (A or B wins) = 1/3 + 1/5 = 8/15

b) P (A or B or C wins) = 1/3 + 1/5 + 1/9 = 29/45

c) P (no one will win) = 1 - P (A or B or C will win) = 1 - 29/45 = 16/45

d) P (neither A nor B will win) = 1 – P(A or B will win)

= 1 - 8/15

= 15.07

Question If X and Y are two independent events, then X and Y' is: (4 points)

Or.X ∩ Y' and X ∩ Y are mutually exclusive events such that;

X = (X ∩ Y') ∪ (X ∩ Y)

P(X) = P(X ∩ Y') + P(X ∩ Y)

P(X ∩ Y') = P(X) – P(X ∩ Y)

  • P(X) – P(X).P(Y) (Since X and Y are independent)

= P(X∩Y')

=> P(X) (1 – P(Y)) = P(X)P(Y')

Thus X and Y' are also independent.

Question If P(A)=2/3, P(B)=1/2 and P(A)B) = 5/6 then find that events A and B are mutually exclusive or not: (2 points)

Or.P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

5/6 = (2/3) + (1/2) − P (A ∩ B)

⇒ P (A ∩ B) = 0

(Video) Probability - Mutually Exclusive Events - Example | Don't Memorise

Thus events A and B are mutually exclusive.

Question Lisa is trying to understand mutually exclusive events using a piece of data. Show how she can find the probability that a die will show 4 or 5? (3 points)

Or.There are a total of 6 faces on a dice, so the total number of outcomes is 6

The probability that a die will show a 4 is given by P(4) = 1/6

The probability that a die will show a 5 is P(5) = 1/6

Using the rule of multiplication, you can know that the probability of getting 4 or 5 is = P(4 or 5).

P(4) + P(5) - P(4 and 5)

= 1/6 + 1/6 – (1/6+1/6)

=2/6 – 2/6 =0

So the probability of getting a 4 or 5 on a roll is zero. These are mutually exclusive events.

Question Dinesh's teacher is teaching him about probability and has given him a deck of 52 cards and asked him to draw a red card or a 6. Calculate the probability of drawing a red card or a 6 (3 points)

Or.The probability of receiving a red card; (R) = 26/52

The probability of getting a 6: P(6) = 4/52

The probability of getting a red and a 6: P(R and 6) = 2/52

P(R or 6) = P(R) + P(6) - P(R e 6)

= (26/52) + (4/52) - (2/52)

= (30-2/52)



Question Kiara noticed her mother trying to take the fish out to clean the tank. She asked her mother, "How many are male and how many are female?" His mother replied that the tank contained 5 male and 8 female fish. What is the probability of drawing the first one, is it a male fish? (4 points)

Or.This question can be solved with the formula

Probability of an event = number of possible outcomes / total number of favorable outcomes

Number of male fish = 5

Number of females = 8

(Video) Probability of Mutually Exclusive Events With Venn Diagrams

total number of fish

5 + 8 = 13

The probability that the fish drawn is a male fish:

Number of male fish / total number of fish

The probability that the fish came from a male fish = 5/13

Question What is the probability of having a king or a queen in a deck at the same time? Is this an example of a mutually exclusive event?(3 points)

Or.In a deck of 52 cards:

  • the probability of a king is 1/13, so P(king)=1/13
  • the probability of a queen is also 1/13, so P(queen)=1/13

The Probability of a KingorA queen uses the formula:

P(A or B) = P(A) + P(B) - P(A and B)

(1/13) + (1/13) – (1/13 + 1/13) =0

The probability is 0. So this is an example of mutually exclusive events.

Question In a language study, a group of 30 people was selected for the study, in which approximately 16 people are learning French and 21 people are learning Spanish. Discover: (5 points)
(EU)It is an example of mutually exclusive events
(ii) The number of people learning both languages
(iii) The probability of Spanish or French


  1. This is not a case of mutual exclusion (there is an opportunity to learn French and Spanish).
  1. We sayBis the number of people learning both languages:
  • Persons studying only French must be 16-b
  • People studying only Spanish must be 21-b

Data available30people whole, then:

(16−b) + b + (21−b) = 30

37 - b = 30

b = 7

So the number of people who speak both languages ​​is 7.

  1. To find out the probability
  • P (French) = 16/30
  • P (Spanish) = 21/30
  • P (French only) = 9/30
  • P (Spanish only) = 14/30
  • P (French and Spanish) = 7/30

Finally, check with our formula:

P(A or B) = P(A) + P(B) − P(A and B)

Enter the values:

P (French or Spanish) = 16/30 + 21/30 − 7/30

= 30/30 = 1

Therefore the value is 1.

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(Video) Mutually Exclusive and Exhaustive Events (6.4)


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