In probability, an event is defined as the set of outcomes obtained from a random experiment. The event example is tossing a coin and there are different types of events. Mutually non-exclusive events are the exact opposite of mutually exclusive events. In non-mutually exclusive events, two events must have at least one common outcome. Refer to the below-mentioned sections to know the definition, examples and mutually non-exclusive events formula.
Definition of Mutually Non-Exclusive Events
Two events are said to be mutually non-exclusive events if both the events have at least one outcome in common between them. Any two events cannot prevent the occurrence of one another so we can say that those events have something common in them. Check the following sections to get examples of non-mutually exclusive events.
Mutually Non-Exclusive Events Examples
Here we are giving some of the examples of mutually non-exclusive events.
- In the case of rolling a die the event of getting an even face and the event of getting less than 5 are not mutually exclusive and they are also known as compatible events.
- While tossing two coins simultaneously the event of getting at least one head and the event of getting no tail are not mutually exclusive.
- Let us take two events A = {2, 3, 5, 7, 8}, B = {1, 2, 4, 6, 8} and A ∩ B = {2, 8}. So, events A and B are not mutually exclusive events.
Addition Theorem on Mutually Non-Exclusive Events
If A and B are two mutually non-exclusive events, then the probability of ‘A union B’ is the difference between the sum of the probability of A and the probability of B and the probability of ‘A intersection B’ is follows. It can also called as the mutually non-exclusive events formula:
P(A U B) = P(A) + P(B) – P(A ∩ B)
The proof for the above statement is here.
Proof: The events A – AB, B – AB are pair wise mutually exclusive events then,
A = (A – AB) + AB,
B = AB + (B – AB)
Now, P(A) = P(A – AB) + P(AB)
P(A – AB) = P(A) – P(AB)
In the same way, P(B – AB) = P(B) – P(AB)
Again, P(A + B) = P(A – AB) + P(AB) + P(B – AB)
P(A + B) = P(A) – P(AB) + P(AB) + P(B) – P(AB)
= P(A) + P(B) – P(AB)
= P(A) + P(B) – P(A) P(B)
Therefore, P(A U B) = P(A) + P(B) – P(A ∩ B)
Solved Problems on Non-Mutually Exclusive Events
Problem 1:
A box has 50 tokens numbers 1 to 50. If the token is drawn at random, what is the probability that the number drawn is a multiple of 2 or 7?
Solution:
Let X be the event of ‘getting a multiple of 2’ and Y be the event of ‘getting a multiple of 7’
The events of getting multiple of 2 (X) = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50}
The total number of multiple of 2 = 25
P(X) = \(\frac { 25 }{ 50 } \) = \(\frac { 1 }{ 2 } \)
The events of getting multiple of 7 (Y) = {7, 14, 21, 28, 35, 42, 49}
The total number of multiple of 7 = 7
P(Y) = \(\frac { 7 }{ 50 } \)
Between the X and Y the favorable outcomes are {14, 28, 42}
The total number of common multiple of both numbers 2 and 7 are 3
The probability of getting a ‘multiple of 2’ and a ‘multiple of 7’ from the numbered 1 to 50 = P(X ∩ Y) = \(\frac { 3 }{ 50 } \)
Therefore, X and Y are non mutually exclusive events.
We have to find out the Probability of X union Y.
So according to the addition theorem for mutually non- exclusive events, we get;
P(X ∪ Y) = P(X) + P(Y) – P(X ∩ Y)
= \(\frac { 1 }{ 2 } \) + \(\frac { 7 }{ 50 } \) – \(\frac { 3 }{ 50 } \)
= \(\frac { 29 }{ 50 } \)
Hence, the probability of getting multiple of 2 and 7 = \(\frac { 29 }{ 50 } \)
Problem 2:
In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?
Solution:
Let girl be the event, A be the event of getting A grade, and boy be the event.
Probability of girl = P(girl) = \(\frac { 13 }{ 30 } \)
Probability of A grade students = P(A) = \(\frac { 9 }{ 30 } \)
Probability of girls getting A grade = P(girl and A) = \(\frac { 5 }{ 30 } \)
Probability of choosing a girl or an A grade student = P(gil U A) = P(girl) + P(A) – P(girl ∩A)
= \(\frac { 13 }{ 30 } \) + \(\frac { 9 }{ 30 } \) – \(\frac { 5 }{ 30 } \)
= \(\frac { 17 }{ 30 } \)
Therefore, the probability of choosing a girl or an A student is \(\frac { 17 }{ 30 } \).
Frequently Asked Question’s
1. What are some examples of non-mutually exclusive events?
Non-mutually exclusive events are events that can occur at the same time. Some of the examples are even numbers and prime numbers on a die, losing a game and scoring points, driving and listening to the radio.
2. How to know if an event is not mutually exclusive?
Two events are called non-mutually exclusive events if they have minimum one common element. So, check for the common elements in both events to say those are mutually non-exclusive events.
3. What does it mean to say something is not mutually exclusive?
In general, mutually exclusive means, one event cannot true if the other is true. Not mutually exclusive means they can takes place simultaneously.