# Independent Events

Published on :

21 Aug, 2024

Blog Author :

N/A

Edited by :

Ashish Kumar Srivastav

Reviewed by :

Dheeraj Vaidya

## What Are Independent Events?

Independent event is a term widely used in statistics, which refers to the set of two events in which the occurrence of one of the events doesn’t impact the occurrence of another event of the set. In other words, these are those events that don’t provide any information about the occurrence or non-occurrence of other events.

For concluding whether the events are dependent or not, one needs to analyze whether the occurrence of one event may alter the probability of the occurrence of the second event. One may calculate the probability of both events and apply multiplication rules to test the independence test.

##### Table of contents

- Independent event is widely used in statistics. In other words, it refers to the set of two events in which one event's happenings do not influence the occurrence of another set event.
- In the probability, two events are independent if the result of one event is not decisive of the probability of occurrence or non-occurrence of another event.
- One can also use it to calculate the probability of both events and apply multiplication rules to test the independence test.

### Independent Events Explained

Independent events are those scenarios whose outcome is not dependent on any other scenario. Moreover, the chances of their appearance are not dependent on other events. The **independent events formula** is closely linked to the probability theory where the chances of occurrence of a particular event affecting the outcome of another event is clearly discussed and evaluated.

In a usual scenario, the occurrence or non-occurrence of a particular event may provide an insight into other events. However, the same is not the case in independent events, since the occurrence or non-occurrence of one event is not going to provide any idea or information about the existence of another event. Thus, the outcome of one of the events is not dependent on the outcome of another event in the same set.

### Examples

The **independent events formula** can be understood in depth with the help of a few examples.

- We take two coins and then toss them. The event of the appearance of a tail or head on one coin is not decisive of the appearance of a tail or head on another coin. Thus, tossing two coins simultaneously or tossing the same coin twice can be said to be independent events. The reason is that the probability of each outcome (i.e., head or tails) is 50% each time and is not dependent on the last toss.
- Similarly, when we take two dice and roll them, the resultant number on one dice does not decide the resultant number on the second dice. As a result, the rolling of two dice is another example.

### Rules

Through the multiplication rule in probability that can be tested to identify whether the two events are independent or not.

Multiplication rules state that, if two events are independent, then:

P(A|B) = P(A)

This mathematical connotation denotes those two events, named A and B, are said to be independent when the probability of event A, given that event B occurs, is equal to the probability of event A. It is because, in the case of independent events statistics, the occurrence or non-occurrence of an event doesn’t decide the occurrence or non-occurrence of another event.

Similarly, the following connotation also holds true.

P(B|A) = P(B)

It means that if A and B are two independent events, the probability of event B, given that event A occurs, is equal to the probability of event B.

Further, there is one more observation that is true for such events.

P(A and B) = P(A) * P(B)

The above equation suggests that if events A and B are independent, the probability of both events occurring is equivalent to the product of their individual probabilities.

### Probability

In the terminology of probability, two events can be said to be independent if the outcome of one event is not decisive of the probability of occurrence or non-occurrence of another event.

Following is the **independent events formula** of probability for any event -

For example, let us calculate the probability of getting 6 on the dice when we roll it. Here, the total number of outcomes is six (numbers 1,2,3,4,5, and 6), and the number of favorable outcomes is one (number 6). Hence, the probability comes out to be 0.16.

### Independent vs. Dependent Events

Independent and dependent events are exactly the opposite effects of each other. Let us understand the dependent and **independent events statistics **through the comparison below.

- Two events are said to be independent when the probability of one event does not impact the probability of another event. For example, simultaneously tossing two coins are independent events because the probability of a head or tail on the first coin is not dependent or decisive of the probability of a head or tail on another coin.
- On the other hand, two events are called dependent if the outcome of one of the events can alter the probability of another event. In simple terms, when the outcome of one event can influence the occurrence of another event, the events are said to be dependent events. For example, in a deck of 52 cards, two cards are chosen randomly one by one. Now, if the first card is chosen and it is not replaced, the probability of the second card will definitely change since after the first card is removed, only 51 cards are to remain in the deck. It results in the two events being dependent events.

### Frequently Asked Questions (FAQs)

**What is the probability of two independent events?**The probability of the simultaneous occurrence of two independent events is equal to the probabilities product. For example, suppose there are two independent events, A and B, then it will give the probability as P(A and B) = P(A) x P(B).

**Are independent events always mutually exclusive?**The difference between mutually exclusive and independent events is that a mutually exclusive event can be called a circumstance if the two events cannot coincide. In contrast, an independent event is when one event remains unaffected by the happening of another event.

**Are independent events conditionally independent?**Independent events must not be conditionally independent. But, there are conditioning events such that independent events are also conditionally independent given. Therefore, one can have two conditionally independent events but not unconditionally independent.

**Can independent events overlap?**There is no effect of an occurrence with another; they are independent. Therefore, the sets will not overlap in the case of mutually exclusive events. However, the sets result of one event does not influence the outcome of the other event. Therefore, they can appear together.

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