Mutually exclusive events are those events that cannot happen at the same time; if one occurs, the other cannot. For example, in a cricket match between India and Pakistan, only one team can win. If Pakistan wins, India loses, and vice versa. Therefore, "India winning" and "Pakistan winning" are mutually exclusive events, as the occurrence of one excludes the other.
Some other examples of mutually exclusive events are,
- Tossing a coin, we either get a head or a tail. Head and tail cannot appear simultaneously. Therefore, the occurrence of a head or a tail are two mutually exclusive events.
- In throwing a die, all 6 faces numbered 1 to 6 are mutually exclusive, as if any one of these faces comes on the top, the possibility of others in the same trial is ruled out.
How to Calculate Mutually Exclusive Events?
We know that mutually exclusive events are events that can not occur simultaneously and if we take two events A and B as mutually exclusive events and the probability of A is P(A) and  the probability of B is P(B), then the probability of happening both events together is, P(A∩B) = 0
Then the probability of any one event is
P(AUB) = P(A) or P(B) = P(A) + P(B)
Here, we define ∩, the symbol, as the intersection of the set and the U symbol as the union of the set. Before proceeding further, let's learn about the intersection of the set and the union of the set.
Intersection of Sets
The symbol that defines the intersection is "∩"; it is also called "AND." We define Intersection as the values that are contained in both sets, i.e.,
If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6}.
Then A intersection B is represented as A∩B and A∩B = {2, 4, 6}
Union of Sets
The symbol that defines the union is "U"; it is also called "OR." We define "union" as all the values contained in both sets, i.e.,
If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6}.
Then A union B is represented as AUB and AUB = {1, 2, 3, 4, 5, 6}
Probability of Mutually Exclusive Events or Disjoint Events
We know that mutually exclusive events can never happen simultaneously, and thus the probability of mutually exclusive events is always zero. We define the probability of the mutually exclusive events A and B as,
P (A∩B) = 0...(i)
We know that,
P (A U B) = P(A) + P(B) - P (A∩B)
But if A and B are mutually exclusive events, then by (i) we get,
P (A U B) = P(A) + P(B)
Example: In a coin toss probability of getting a head is P(H) and the probability of getting a tail is P(T) and both getting head and getting a tale is a mutually exclusive events,
Then,
- P(H) = 0.50
- P(T) = 0.50
P(H∩T) = 0
P(HUT) = P(H) + P(T) = 0.50 + 0.50 = 1
Venn Diagram of Mutually Exclusive Events
Venn diagrams are widely used to represent mutually exclusive events. We know that if we represent a set using a circle in the Venn diagram, then in mutually exclusive events we get the Venn diagram in which we have nothing in common between the two sets, as shown in the image below:
Non-Mutually Exclusive Events
In non-mutually exclusive events we get the Venn diagram in which we have some common parts between the two sets as shown in the image below:
Do Mutually Exclusive Events Add up to 1?
We know that it is not possible for mutually exclusive events to occur simultaneously. The probability of any event can never be greater than one, and as we already know, mutually exclusive events are dependent events, and thus their probability is never greater than one.
Now, the probability of mutually exclusive events can add up to 1 only if the events are exhaustive, i.e., at least one of the events is true.
Mutually Exclusive Events Probability Rules
We use the following rules for simplifying the mutually exclusive events, such as if A and b are two mutually exclusive events, then
- Addition Rule: P (A + B) = 1
- Subtraction Rule: P (A U B)’ = 0
- Multiplication Rule: P (A ∩ B) = 0
Simple events that only have one possible outcome are always mutually exclusive to other simple events.
Conditional Probability for Mutually Exclusive Events
Conditional probability is the probability of event A occurring given that event B has already occurred. We define the conditional probability of event B when event A has already occurred as P(B|A). We can calculate its value as
P(B|A) = P(A ∩ B)/P(A)
Now if A and B are two mutually exclusive events, then by using the multiplication rule, P (A ∩ B) = 0.
P(B|A) = 0/P(A) = 0
Thus, the formula for conditional probability for mutually exclusive events is,
P(B | A) = 0
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Solved Examples
Example 1: If P(A) = 0.20, P(A ∪ B) = 0.51, and A and B are mutually exclusive events, then find P(B).
Solution:
Given,
P(A ∪ B) = 0.51 and P(A) = 0.20We know that,
P(A ∪ B) = P(A) + P(B)
0.51 = 0.20 + P(B)
P(B) = 0.51 - 0.20 = 0.31Thus, P(B) is 0.31
Example 2: In a deck of 52 cards, find the probability of getting either an even card or a face card.
Solution:
Probability of getting a even card P(A) = 5/13
Probability of getting a face card P(B) = 3/13We know that P(A) and P(B) are two mutually exclusive events, then
P(A ∪ B) = P(A) + P(B)
= 5/13 + 3/13
= 8/13Thus, the probability of getting either a even card or a face card is 8/13.
Example 3: If P(B) = 0.35, P(A ∪ B) = 0.65, and A and B are mutually exclusive events, then find P(A).
Solution:
Given,
P(A ∪ B) = 0.65 and P(B) = 0.35We know that,
P(A ∪ B) = P(A) + P(B)
0.65 = 0.65 + P(A)
P(A) = 0.65 - 0.35 = 0.30Thus, P(A) is 0.30