Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Difference Between Mutually Exclusive and Independent Events

ffImage
Last updated date: 17th Apr 2024
Total views: 34.8k
Views today: 0.34k
hightlight icon
highlight icon
highlight icon
share icon
copy icon

Mutually Exclusive Events

The two events A and B are said to be incompatible if both events do not occur at the same time. For example, consider the throwing of a coin. Let A be the event where the coin rests on the heads and let B be the event where the coin sits on the tails. It follows that, in a single throw of the right coin, events A and B are not the same. Collaborative variations can be shown in the Venn diagram (read Venn Diagrams). This Venn diagram depicts two special events A and B.


It is not possible for events A and B to occur simultaneously


(Image Will be Updated Soon)


Independent Events

Events A and B are said to be independent if the chances of B happening are not affected by the occurrence of event A. For example, suppose we throw a coin twice. Let A be the event where the first coin throws the world to the head. In addition, B should be an event where the second coin throws the earth to the heads. Obviously, the effect of the first coin toss does not affect the effect of the second coin toss. See Tree Pictures for this example in detail. The A and B are independent even as follows.


Consider the illustration of the tree shown here. The probability that incident R occurred near the second branch depends on the outcome of the first incident. The events B and R are not independent. Therefore, they are known as dependent events and related opportunities are conditional.


(Image Will be Updated Soon)


What are Mutually Exclusive Events?

Get ready to know the difference between Mutually exclusive and independent events and also to know what mutually exclusive and independent events are!

  • Two events let’s suppose event A and event B are said to be mutually exclusive if it is not possible that both of the events (A and B) occur at the same time.

  •  For example, let’s consider the toss of a coin. When we toss a coin let A be the event that the coin lands on heads and let B be the event that the coin lands on tails. 

  • In a single fair coin toss, events A and B are mutually exclusive which means the outcome can be either tails or heads.  We cannot get both heads and tails at the same time.

  •  Mutually exclusive events can be represented using a Venn diagram.

The following Venn diagram given below shows two mutually exclusive events A and B:

If event A occurs, then there is no possibility of the occurrence of event B.


Examples of Mutually Exclusive Events:

There are 52 Cards in a deck:

  • the probability of getting a King = 1/13, so we can say P(King)=1/13

  • the probability of getting a Queen is = 1/13, so we can say P(Queen)=1/13

 When we combine those two Events, we cannot get queen and king at the same time thus,

 P (A and B ) = 0

Therefore, we can say the probability of a King OR a Queen is (1/13) + (1/13) = 2/13


What are Independent Events?

  • Events A and B are known as independent events if the probability of B occurring is unaffected by the occurrence of the event A happening

  • For example, let’s suppose that we are tossing a coin twice. Let A be the event that the first coin toss lands on heads and let B be the event that the second coin toss lands on heads.

  •  Here, the Occurrence of event A does not affect event B in any manner.

  • Independent events can be represented using a Venn diagram.

The following Venn diagram given below shows two independent events A and B:


Formulas of Mutually Exclusive Events and Independent Events!

  1. Probability of any event = Number of favorable outcomes / Total number of outcomes

  2. For mutually exclusive events  = P(A or B) which can also be written as P(A∪B)

= P(A)+P(B)

And here P(A and B ) = 0

  1. For independent events = P(A∩ B) = P(A). P(B)


Difference Between Mutually Exclusive and Independent Event:

At first the definitions of mutually exclusive events and independent events may sound similar to you. The biggest difference between the two types of events is that mutually exclusive basically means that if one event happens, then the other events cannot happen.


Mutually Exclusive and Independent Events

On the other hand, if the events are independent, then it means the occurrence and the outcome of any one event won’t have any effect on the occurrence and outcome of the other events.


Mutually exclusive and Independent events Difference

When an event is sometimes in two cases they are called special events.

If the occurrence of one incident does not control the occurrence of another then it is called an independent event.

The non-event will end with the event 

There is no impact of the incident on each other and they are independent of each other.

The mathematical formula for special events equally can be represented by P (X and Y) = 0

The independent mathematical formula can be defined as P (X and Y) = P (X) P (Y)

Sets will not overlap when there are special events.

Sets will pass if there are private events.


Mutually Exclusive vs Independent Events Examples

  • Outcomes of rolling a die are mutually exclusive events. You can get either 5 or 6, but you can never get 5 and 6 at the same time. 

  • Outcomes of rolling a die two times are independent events. The number we get on the first roll on the die has no effect on the number we’ll get when we roll the die one more time. 

Mutually exclusive and independent events can be differentiated on the basis of Definition, Dependency,  Occurrence of both events, and Venn Diagrams.


Difference Between Mutually Exclusive Event and Independent Event

Comparison on the Basis of

Mutually Exclusive Event

Independent Event

1. Definition

Here the events cannot happen simultaneously.

The occurrence and outcome of one event don’t affect the occurrence and outcome of the other event. 

2. Dependency

Occurrence of event A results in non-occurrence of event B.

Occurrence of event A does not affect event B in any manner.

3. Occurrence of both events

The mathematical formula for the representation of the mutually exclusive event is P(A∩B) = 0

The mathematical formula for the representation of the mutually exclusive event is P(A∩B) = P(A).P(B)

4. Venn Diagram Representation


(Image to be Added Soon)


Here the events A and B do not overlap.


(Image to be Added Soon)


Here the events do overlap.


The following are the key distinctions between mutually exclusive and independent events:

  • Mutually exclusive events occur when two or more things happen at the same time. Independent events occur when the occurrence of one event has no bearing on the occurrence of another.

  • The occurrence of one event will result in the non-occurrence of the other in mutually exclusive events. In independent events, on the other hand, the occurrence of one event has no bearing on the occurrence of the other.

  • P(A and B) = 0 represents mutually exclusive events, while P (A and B) = P(A) P(A)

  • The sets in a Venn diagram do not overlap each other in the case of mutually exclusive events, but they do overlap in the case of independent events.


Questions to Be Solved:

Question 1. If we throw a dice twice, then find the probability of getting two 5’s.

Solution Let’s find the probability of getting 5’s,

The formula for finding the probability is,

Probability=Favorable outcomes/Total possible outcomes.

Total possible outcomes when we throw a dice are 6.

Probability of getting 5 on the first throw = 1/6

Probability of getting 5 on the second throw is also = 1/6

Let’s find the probability (Getting two 5’s), since they are independent events,

Formula: P(A∩B) = P(A). P(B)

Probability of getting two 5’s = 1/6 ×1/6

Therefore, Probability of getting two 5’s = 1/6 ×1/6 = 1/36

FAQs on Difference Between Mutually Exclusive and Independent Events

1. Are Independent Events Always Mutually Exclusive and Are Mutually Exclusive Events Independent or Dependent?

No, mutually exclusive events (the events with non-zero probability) are always dependent. The definition of independence for events R and Q says that P(R and Q) = P(R) P (Q). However, if events P and Q are mutually exclusive, then P(R and Q) is equal to zero. If events R and Q are independent, then having information on event R does not tell us anything about event Q. If events R and Q are disjoint, then knowing that R occurs gives us information that event Q cannot occur. Disjoint or mutually exclusive events are always dependent since if one event occurs we already know the other one cannot happen.

2. What are Independent and Dependent Events and How Do You Know if an Event is Independent?

An event in which the outcome isn't affected by another event is known as an independent event. A dependent event is an event that is affected by the outcome of a second event. To test whether any two given events suppose  A and B are independent, then we need to calculate P(A), P(B), and P(A∩B), and then we need to check whether P(A∩B) equals P(A)P(B). If they are equal, then events A and B are independent; if not, events A and B are dependent.

3. How to identify if events are independent?

Events A and B are independent if the number P (A∩B) = P (A) · P (B) is accurate. You can use equations to check whether events are independent; multiply the possibilities of the two events together to see if they are equal to the probability that both events occur together.

4. What is the difference between inclusive and special events?

2 events are not the same where both cannot occur at the same time. 2 events combine together if both can occur at the same time.

5. How to explain 2 different events?

Special Events Partnerships

  • Turn left and turn right It is Mutually Exclusive (you can't do both at once)

  • Coin Throwing: Heads and Tails Separated.

  • Cards: Kings and Special Aces.

6. How do you know if A and B are alone?

A and B are special events if they do not occur at the same time. This means that A and B do not participate in any of the results and P (A NO-B) = 0. and is not equal to zero.

7. What are PA and B?

Terms with conditions: p (A | B) probability of event A, in the case of B. ... Combined opportunities: p (A and B). Opportunities for event A and event B to occur. The possibility of a collision of two or more instances. The probability of a combination of A and B may be marked p (A ∩ B).