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Mutually exclusive events is a term extensively used in probability. It refers to two events which cannot happen simultaneously. For instance, a laptop can be either in the condition of on or off, it cannot be on and off at the same time, isn’t it?. So the ‘on’ state is one event and the ‘off’ state is another event. So, here, this is an example of mutually exclusive events in which the event ‘on’ prevents the ‘off’ event in the laptop and vice versa. One can also infer from this example that in mutually exclusive events one event prevents the other from happening.
To understand about Mutually exclusive events, one must be familiar with the basic concept of probability.
Read more: Types of events
Key Terms: Mutually exclusive events, disjoint events, Probability, Dependent events, independents events, conditional probability,
Mutually Exclusive Events Definition
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Any two events which cannot occur at the same time are termed as Mutually Exclusive Events with relation to each other. Such events are also called disjoint events. Mutually exclusive events always have different outcomes in relation to each other. Mathematically, mutually exclusive events are those in which the probability of the events, in concern, is zero or without any value.
How to find out and show that two events are Mutually exclusive events?
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In mathematical probability theorem, if ‘A’ and ‘B’ are two mutually exclusive events then their probability can be written as P(A ∩ B) Or P(A Or B). This symbol ∩ means ‘and’ and the Probability of occurrence of event ‘W’ and event ‘X’ would be zero. Thus,
P(W ∩ X) = 0
The probability of two events ‘A’ and ‘B’ is calculated in the following way:
P(A ∩ B) = P(A) + P(B)
If the result of P(A) + P(B) = 0 Then, these two are mutually exclusive events.
For events which are not mutually exclusive events, the probability P (A ∪ B) is calculated by the following formula:
P (A ∪ B) = P(A) + P(B) – P (A and B)
Read more:
| Relevant Concepts | ||
|---|---|---|
| Difference between mutually exclusive events & independent events | Probability & statistics | Chance & probability |
| Combinatorics | Multiplication rule of probability | Set Operations |
Mutually Exclusive Events Rules and formula
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As already mentioned before, mutually exclusive events do not occur at the same time. This doesn’t imply that mutually exclusive events are collectively exhaustive.
For example, a traffic signal which shows only green and red light. At any instance the signal light will be either green or red, but not both. So, these are mutually exclusive events. Moreover, there’s no possibility of a third outcome. So, these are collectively exhaustive events. Collectively exhaustive possibilities are those in which there is no chance of some other result or probability.
Traffic signal indicating red and green signal
For instance, Imagine throwing a Ludo dice, the events of getting the numbers 1 and 3 are mutually exclusive in nature as both can’t happen together. But these two possibilities are not collectively exhaustive as there is a good chance of getting a third possible outcome (2,4,5 or 6).
From the above definition of mutually exclusive events, it can be concluded certain rules for any two probabilities related to each
Addition Rule - P( X + Y) = 1
Subtraction rule - P( X U Y) = 0
Multiplication rule - P( X ∩ Y) = 0
With certain alterations in the condition involved there are a variety of events.
For example, a coin that has a Head on both the sides of the coin or a Tail on both sides. It doesn’t matter how many times one flips it, it will always occur Head (for the first coin) and Tail (for the second coin).
If the sample space of such an experiment is checked, it will always be Head for the first coin and Tail for the second one. Such events have single points in the sample space and are called “Simple Events”. Such kinds of two sample events are always mutually exclusive to each other.
Mutually Exclusive Events: Formula
These mutually exclusive events formulas can be used to solve the questions based on mutually exclusive events probability.
The probability of two events says A and B are mutually exclusive is represented as
- A and B
The intersection set between A and B is equal to {null}. Hence P (A and B) = 0.
It is because when two events cannot happen at the same time, then obviously there will be nothing common in that.
- 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 the addition of the probabilities of the events separately.
Dependent and Independents Events
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Two events are categorized as dependents events when one event affects the probability of the occurrence of the other event. On the other hand, two events which do not affect the probability of each other in any way are termed as independent events. Mutually Exclusive are Independent Events as they affect the probability of each other. Moreover, independent events can never be mutually exclusive.
Read more: Events in Probability
Examples of Mutually Exclusive Events
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- A person can either turn left or right at the same time. So it is an example of mutually exclusive events.
- In a deck of cards Kings and Aces are mutually exclusive events as one can pull either an Aces card or a King card at a time.
- Occurrence of day and night is another excellent example by nature which are mutually exclusive to each other.
- A gas stove cannot be on and off at the same time so these are mutually exclusive events.
Conditional Probability for Mutually Exclusive Events
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Another special case of probability is conditional probability in which if one event happens then only the other event takes place or else not. Thus, the second event is based on the condition of occurrence of the first one. Such events are termed as conditional events.
Conditional Probability for two independent events ‘A’ and ‘B’ in which event B has given event A is denoted by the expression P( B|A) and it is defined by the following equation:
P(B|A)= P (A ∩ B)/P(A)
The above equation using multiplication rule becomes: P (A ∩ B) = 0
P(B|A)= 0/P(A)
So the conditional probability for mutually exclusive events is:
P (B | A) = 0
Things to remember
- Mutually exclusive events are those events which cannot happen at the same time with relation to each other. They are also known as disjoint events.
- Mutually exclusive events are dependent events, as they affect the happening of each other.
- We can demonstrate that two given events A and B are mutually exclusive in nature by proving that their probability P(A ∩ B) Or P(A Or B) is zero.
- The probability of the union of two mutually exclusive events is derived by the addition of the probabilities of the two events separately.
Sample Questions
Ques. Out of the given set of events state which one is mutually exclusive: (4 marks)
(i) Pulling out a King from a deck of cards and a heart card.
(i) Getting a 1 and 3 when a dice is thrown.
(iii) Getting a tail from a coin with Tail on both sides.
(iv) Getting a 3 and 4 on throwing a dice.
Ans.
- It is not an example of a mutually exclusive event, because in a deck of cards there is a card of ‘King of Hearts’ which has both King and Heart, so getting a King and Heart card at the same time is possible. Hence, it is not a mutually exclusive event.
- This is an example of a mutually exclusive event. When a dice is thrown one cannot get 1 and 3 at the same time.
- It is not a mutually exclusive event, because the coin has Tail on both sides and only the event is mentioned in the question. Without a second relatable event it cannot be decided if it is a mutually exclusive event or not. It is a simple event.
- This is a mutually exclusive event. Here, there is no possibility of getting a 1 and 3 at the same time.
Ques. Three coins are tossed at the same time. We say A as the event of receiving at least 2 heads. Likewise, B denotes the event of getting one and C is the event of getting heads on the second coin. Which of these is mutually exclusive? (5 marks)
Ans. Firstly, create a sample space for each event. For the event ‘A’ we have to get at least two heads. Therefore, include all the events that have two or more heads.
Or can be written as:
A = {HHT, HTH, THH, HHH}.
This set A contains 4 elements or events in it i.e. n(A) = 4
In a similar way, for event B, can be written as:
B = {TTT} and n(B) = 1
This set has only one element.
In the same way,
C = {THT, HHH, HHT, THH} and n(C) = 4
So B & C and A & B are mutually exclusive as they have nothing in their intersection.
Ques. The chances of the 3 teams a, b, c winning a football match are 1 / 3, 1 / 5 and 1 / 9 respectively. Find the probability that (3 marks)
a] out of the three teams, either team a or team b will win
b] either team a or team b or team c will win
c] none of the teams will win the match
d] neither team a nor team b will win the match
Ans. a) P (A or B will win) = 1/3 + 1/5 = 8/15
b) P (A or B or C will win) = 1/3 + 1/5 + 1/9 = 29/45
c) P (none 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(either A or B will win)
= 1 – 8/15
= 7/15
Ques. If X and Y are two independent events, then X and Y’ is: (4 marks)
Ans. 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.
Ques. If P (A) = 2 / 3, P (B) = 1 / 2 and P (A ∪ B) = 5 / 6 then find out that if events A and B are mutually exclusive or not: (2 marks)
Ans. P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
5 / 6 = (2 / 3) + (1 / 2) − P (A ∩ B)
⇒ P (A ∩ B) = 0
Thus the events A and B are mutually exclusive.
Ques. Lisa is trying to understand mutually exclusive events with the help of a dice. Show how she can find out the probability of a dice showing 4 or 5? (3 marks)
Ans. There are a total of 6 faces on a dice, hence, the total number of outcomes will be 6
The probability of a dice showing 4 is given by P(4) = 1/6
The probability of a dice showing 5 is P(5) = 1/6
Using the multiplication rule can know 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
Thus the probability of getting 4 or 5 in one throw is zero. These are mutually exclusive events.
Ques. Dinesh's teacher is teaching him about Probability and gave him a deck of 52 cards and asked him to select a red card or a 6. Work out the probability of selecting a red card or a 6 (3 marks)
Ans. The probability of getting a Red card; (R) = 26/52
The probability of getting a 6: P(6) = 4/52
The probability of getting both a Red and a 6 : P(R and 6)= 2/52
P(R or 6) = P(R) + P(6) - P(R and 6)
= (26/52) + (4/52) - (2/52)
= (30-2/52)
=28/52
=7/13
Ques. Kiara noticed her mother trying to take out the fish to clean the fish tank. She asked her mother, "How many are males and how many are females?" Her mother replied that the tank contained 5 male fish and 8 female fish. What is the probability of taking out the first, is a male fish? (4 marks)
Ans. This question can be solved by using the formula.
Probability of an event = Number of possible outcomes/ Total no of favorable outcomes
No. of male fish = 5
No. of female fish = 8
Total no of fishes
5+8 = 13
The probability that the fish taken out is a male fish:
Number of male fish/ Total number of fish
The probability that the fish are taken out a male fish = 5/13
Ques. What is the probability of getting a King or a Queen at the same time in a deck of cards? Is this an example of a mutually exclusive event? (3 marks)
Ans. 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 King or a Queen is by using 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. Hence, this is an example of mutually exclusive events.
Ques. In a language study a group of 30 people was selected for the study in which about 16 people study French, 21 study Spanish. Find out: (5 marks)
(i) Is it an example of mutually exclusive events
(ii) The number of people studying both the languages
(iii) The probability of Spanish or French
Ans.
- This is not a case of Mutually Exclusive (there is a possibility that one can study French and Spanish).
- Let's say b is the number of people studying both languages:
- people studying French Only must be 16-b
- people studying Spanish Only must be 21-b
Given: There are 30 people altogether, so:
(16−b) + b + (21−b) = 30
37 − b = 30
b = 7
so, the number of people speaking both languages is 7.
- To find out the probability of
- P(French) = 16/30
- P(Spanish) = 21/30
- P(French Only) = 9/30
- P(Spanish Only) = 14/30
- P(French and Spanish) = 7/30
Lastly, check with our formula:
P(A or B) = P(A) + P(B) − P(A and B)
Put the values in:
P(French or Spanish) = 16/30 + 21/30 − 7/30
= 30/30= 1
Hence, the value is 1.
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