What are Mutually Exclusive Events in Probability

An event is an outcome or a combination of outcomes of an experiment. Suppose we throw a die. Let E be the event of a perfect square number, then E = {1,4} is the answer. Whenever an outcome satisfies the conditions, given in the event, we say that the event has occurred.

The definition of mutually exclusive events can also be extended to more than two events. More than two events are mutually exclusive, if the happening of one of these, rules out the happening of all other events. The events A = {1, 2}, B = {3} and C = {6}, are mutually exclusive in connection with the experiment of throwing a single die. In this section, we will study about mutually exclusive events in detail.

There are different types of events in probability we deal with. Some of them are,

  • Simple Event
  • Compound Event
  • Null Event
  • Sure Event or Certain Event
  • Complement of an Event
  • Mutually Exclusive Events
  • Exhaustive Events

How to Define Mutually Exclusive Events

When two events associated with a random experiment are said to be mutually exclusive if both cannot occur together in the same trial. Mutually-exclusive events, also known as disjoint events.

It is also important to distinguish between independent and mutually exclusive events. Independent events are those which do not depend on one another; while mutually exclusive events cannot occur together at one time.

For example: In the experiment of throwing a die, the events A = {1, 4} and B = {2, 5, 6} are mutually exclusive events.

In the same experiment, the events A = {1, 4} and C = {2, 4, 5, 6} are not mutually exclusive because, if 4 appears on the die, then it is favorable to both events A and C.

If A and B are two events, then A or B or (A u B) denotes the event of the occurrence of at least one of the events A or B.

A and B or (A n B) is the event of the occurrence of both events A and B.

If A and B happen to be mutually exclusive events, then P(A n B) = 0.

Important Result:

The probability of one or other events is equal to the sum of their separate probabilities.

For two events X and Y, we have

P(XY)=P(X)+P(Y)P(XY)P (X \cup Y) = P(X) + P(Y) – P(X \cap Y)\\

If in case X and Y are mutually exclusive events, then there will be no common event.

So, P(XY)=0P(X \cap Y) = 0

Therefore, P(XY)=P(X)+P(Y)P (X \cup Y) = P(X) + P(Y)\\

Related articles

Bayes theorem of probability

Dependent events

Probability JEE Main Previous Year Questions

Solved Examples on Mutually Exclusive Events

Below are a few examples to understand the concept in a better way:

Example 1: A pair of dice is rolled. Find the probability

(i) getting either even numbers or odd numbers.

(ii) the sum of the numbers rolled is either 6 or 10.

Solution:

(i) Possible outcomes for even numbers: (2, 2), (4, 4), (6, 6)

=> P(even numbers) = 3/36 = 1/12

Possible outcomes for odd numbers:  (1, 1), (3, 3), (5, 5)

=> P(Odd numbers) = 1/12

P (even or Odd) = 1/12 + 1/12 = 2/12 = 1/6

(ii)

(1, 5), (2, 4), (3, 3), (4, 2), (5, 1) -> (5 outcomes that have sum 6)

(6, 4), (5, 5), (4, 6) -> (3 outcomes that have sum 10)

Now,

Probability of 6: P(6) = 5/36

Probability of 10: P(10) = 3/36

Both the events are mutually exclusive since the sum of numbers cannot be 6 and 10 at the same time.

P(6 or 10) = P(6) + P(10) = 5/36 + 3/36 = 8/36 = 2/9.

Example 2: Which of these are mutually exclusive events.

(i) On a throw of a die, “getting 1” and “getting 5”

(ii) Getting “a head” or “a tail”

(iii) Choose “a king” or “a queen” form a deck of cards

(iv) “having an ace” and “having a spade” from a deck of cards.

Solution:

i) On a throw of a die, the two events “getting 1” and “getting 5” are two mutually-exclusive events because we will never get 1 and 5 both at one time in a throw.

ii) Getting a head or a tail are two mutually-exclusive events.

iii) Drawing a king or a queen are mutually-exclusive events because both cannot be drawn in one time.

(iv) The two events “having an ace” and “having a spade” are not mutually exclusive since we may even draw an “ace of spade”. So, these two events can occur in the same draw.

Example 3: The probabilities of three mutually exclusive events are 2/3, 1/4 and 1/6 respectively. Verify whether the statement is correct?

Solution:

Let the events be A, B, and C.

If the events are mutually exclusive then, AB=ΘA\cap B=\Theta BC=ΘB\cap C=\Theta and AC=Θ A\cap C=\Theta .

So, ABC=ΘA\cap B \cap C=\Theta .

If the above conditions are satisfied, then P(ABC)=P(A)+P(B)+P(C)P(A\cup B\cup C)=P(A)+P(B)+P(C)

Since P(ABC)=1312>1P(A\cup B\cup C)=\frac{13}{12}> 1, the probability value lies within 1.

Therefore the statement is wrong.

Example 4: If P (A) = 1/3, P (B) = 2/3, then check whether

a] A & B are mutually exclusive.

b] A & B are exhaustive.

Solution:

The events are said to be mutually exclusive if P(AB)=0P(A\cap B)=0.

The events are exhaustive if P(AB)=1P(A\cup B)=1.

P(A)+P(B)=23+13=1P(AB)=P(A)+P(B)P(AB)1=10P(A)+P(B)=\frac{2}{3}+\frac{1}{3}=1\\ P(A\cup B)=P(A)+P(B)-P(A\cap B)\\1=1-0

If [AB][A\cup B] be the sample space, then the above two conditions are true.

Hence A and B are mutually exclusive and exhaustive.

Example 5: Events A, B, C are mutually exclusive events such that P(A)=3x+13,P(B)=1x4,P(C)=12x2P(A)=\frac{3x+1}{3}, P(B)=\frac{1-x}{4},P(C)=\frac{1-2x}{2}. Then find the set of all possible values of x are in the interval.

Solution:

Given that the events are mutually exclusive.

P(A)0,P(B)0,P(C)0, and P(ABC)0P(A)0,P(B)0,P(C)0, and P(A)+P(B)+P(C)03x+130,1x4>0,12x203x+13+1x4+12x20x>13,x1,x12,[13x]>013x12 and x1313x12xϵ[13,12]P(A)\geq 0,P(B)\geq 0,P(C)\geq 0, \text \ and \ P(A\cup B\cup C)\geq 0\\ \Rightarrow P(A)\geq 0,P(B)\geq 0,P(C)\geq 0, \text \ and \ P(A)+P(B)+P(C)\geq 0\\ \frac{3x+1}{3}\geq 0,\frac{1-x}{4}>0, \frac{1-2x}{2}\geq 0\\ \frac{3x+1}{3}+\frac{1-x}{4}+ \frac{1-2x}{2}\geq 0\\ x > \frac{-1}{3}, x \leq 1, x \leq \frac{1}{2}, [1-3x] > 0\\ \frac{-1}{3} \leq x \leq \frac{1}{2} \text \ and \ x\leq \frac{1}{3}\\ \frac{-1}{3} \leq x \leq \frac{1}{2}\Rightarrow x\epsilon [\frac{-1}{3},\frac{1}{2}]

Example 6: Two dice are thrown and the sum of the numbers which come up on the dice is noted. Consider following events associated with this experiment.

A : The sum is less than or equal to 3

B: The sum is greater than 11.

Check whether these pair of events are mutually exclusive.

Solution:

Number of elements in the sample space S = 36

Then A = {(1,1), (1,2), (2,1)}

B = {(6,6)}

A⋂ B = Ø

Hence A and B are mutually exclusive events. 

Example 7: A coin is tossed three times, consider the following events.

A: No tail appears

 B: Exactly one tail appears

Do they form a set of mutually exclusive events?

Solution:

The sample space S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

A = {HHH}

B = { THH, HTH, HHT}

A⋂ B = Ø

Hence A and B are mutually exclusive events.