Non-Mutually Exclusive Events
Last Updated :
25 Oct, 2024
In probability, an event is defined as a specific outcome or a set of outcomes from a random experiment. An event can consist of one or more possible results from the sample space (which is the set of all possible outcomes). In probability there are many different types of Events:
In this article, we will discuss Non-Mutually Exclusive Events along with its definition, conditions for Non-Mutually Exclusive Events, and their examples.
Non-Mutually Exclusive Events
Non-mutually exclusive events also referred to as mutually inclusive events are events that can occur at the same time.
In other words, Non-mutually exclusive events are events that can occur simultaneously, meaning the occurrence of one event does not prevent the occurrence of the other. These events have overlapping outcomes, where both events can happen at the same time.
Some examples of non-mutually exclusive events include:
- Getting a red card and getting a king.
- Getting an odd number and multiple of 3 in a dice throw.
Condition for Non-Mutually Exclusive Events
The condition for two events A and B to be non-mutually exclusive is the intersection of both events should not be empty or zero or ϕ. It is mathematically written as:
A ∩ B ≠ ϕ
In simple words we can say that A nd B are Non-Mutually Exclusive Events if both events can occur together in a single experiment or trial.
Probability of Non-mutually Exclusive Events
The formula for probability occurring of one of two non-mutually exclusive events X or Y is given by:
P (X ∪ Y) = P(X) + P(Y) - P (X ∩ Y)
Where,
- P (X ∪ Y) is probability of occurrence of event X or event Y
- P(X) is probability of occurrence of event X
- P(Y) is probability of occurrence of event Y
- P (X ∩ Y) is probability of occurrence of event X and event Y
Conditional Probability for Non-Mutually Exclusive Events
Conditional probability refers to the probability of occurrence of an event when another event has already occurred. The conditional probability for occurring of B when A has already occurred and events A and B non-mutually exclusive events is:
P (B | A) = P (A ∩ B) / P(A)
where,
- P (B | A) is probability of occurrence of event B when Event A has already occurred.
- P(A) is probability of occurrence of event A
- P (A ∩ B) is probability of occurrence of event A and event B simultaneously.
Venn Diagram of Non-Mutually Exclusive Events
In Venn diagram of non-mutually exclusive events we have some common part between the sets as there is an overlap between the two events. The Venn diagram for two non-mutually exclusive events is represented below:
Non-Mutually Exclusive EventsDifference Between Non-Mutually Exclusive and Mutually Exclusive Events
The table below represents the difference between non-mutually exclusive and mutually exclusive events.
Characteristics | Non-Mutually Exclusive Events | Mutually Exclusive Events |
|---|
Definition | When two events happen simultaneously. | When two events cannot happen simultaneously. |
|---|
Also known as | Mutually inclusive events. | Disjoint events. |
|---|
Condition | A ∩ B ≠ 0 | A ∩ B = 0 |
|---|
Example | Drawing a red and diamond card. | Drawing a red and spade card. |
|---|
Probability Formula | P (X ∪ Y) = P(X) + P(Y) - P (X ∩ Y) | P (X ∪ Y) = P(X) + P(Y) |
|---|
Conditional Probability Formula | P (B | A) = P (A ∩ B) / P(A) | P (B | A) = 0 |
|---|
Solved Examples on Non-Mutually Exclusive Events
Example 1: Given that P(A) = 0.6, P(B) = 0.8 and P (A ∩ B) = 0.5 then find P (A ∪ B) when A and B are Non-Mutually Exclusive Events.
Solution:
To find P (A ∪ B) we use formula:
P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) = 0.6 + 0.8 - 0.5
⇒ P (A ∪ B) = 1.4 - 0.5
⇒ P (A ∪ B) = 0.9
Example 2: For two Non-Mutually Exclusive Events A and B P(A) = 0.5, P(B) = 0.7 and P (A ∪ B) = 0.6 then find P (A ∩ B).
Solution:
To find P (A ∩ B) we use formula:
P (A ∩ B) = P(A) + P(B) - P (A ∪ B)
⇒ P (A ∩ B) = 0.5 + 0.7 - 0.6
⇒ P (A ∩ B) = 1.2 - 0.6
⇒ P (A ∩ B) = 0.6
Example 3: Find the conditional probability P (B|A) given that P (A ∩ B) = 0.4 and P(A) = 0.8 where A and B are Non-Mutually Exclusive Events
Solution:
To find conditional probability P (B|A) we use formula:
P (B | A) = P (A ∩ B) / P(A)
⇒ P (B | A) = 0.4 / 0.8
⇒ P (B | A) = 0.5
Example 4: In a deck of 52 cards find the probability of getting a black card or king.
Solution:
Let P(B) is probability of black cards,
P(K) is probability of king cards
As B and k are Non-Mutually Exclusive Events
P (B ∩ K) is probability of black king card
- P(B) = 26/52 = 1/ 2
- P(K) = 4 / 52 = 1 / 13
- P (B ∩ K) = 2 / 26 = 1/13
Thus, P (B ∪ K) = P(B) + P(K) - P (B ∩ K)
⇒ P (B ∪ K) = 1/2 + 1/13 - 1/ 13
⇒ P (B ∪ K) = 1/2
Practice Questions on Non-Mutually Exclusive Events
You can download free worksheet on Non-Mutually Exclusive Events for practicing various different questions with their answers from below:
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