Probability of Independent Events

JEE CRASH  Lite COURSE 2022

An Introduction to Probability of Independent Events

The event of an occurrence which does not depend on any other event is called an Independent event. For example, if a coin flipped in the air and got the outcome as Head, then again flipped the coin and got the outcome as Tail. In both cases, the occurrence of both events does not depend on each other. It is one of the events in probability. Here is the complete definition of independent events in probability.

Independent Events

In Probability, the set of outcomes of an experiment or an activity is called events. There are various types of events such as independent events, dependent events, mutually exclusive events, etc.


If the probability of occurrence of event A is not dependent on the occurrence of another event B, then A and B are said to be independent events.


Consider an example of rolling a die. If A is the event, where 'the number appearing is odd’ and B is another event, where  ‘the number appearing is a multiple of 3’, then


P(A ∩ B) = \[\frac {1} {6}\]


P(A│B) = \[\frac {P(A∩B)} {P(B)}\]


= \[(\frac {1} {6})\] / \[(\frac {1} {3})\] 


= \[\frac {1} {2}\]


P(A) = P(A│B) = \[\frac {1} {2}\], which proves that event B has no relationship with the occurrence of event A.


Using the Multiplication rule of probability, P(A ∩ B) = P(B). P(A│B)


P(A ∩ B) = P(B) .P(A)

What are Mutually Exclusive Events?

An event is said to be mutually exclusive if both events cannot occur at the same time. Mutually exclusive events do not have an outcome in common.

Independent Events vs Mutually Exclusive Events

Independent Events

Mutually Exclusive Events

The occurrence of the event cannot be specified based on the previous occurrence.

The occurrence can be predicted based on the previous occurrence.

It can have a common outcome.

It cannot have a common outcome.

If A and B are two independent events, then

P(A ∩ B) = P(B) .P(A)

If A and B are two mutually exclusive events, then

P(A ∩ B) = 0

Solved Examples

Example 1:

P(B∩C) + P(C∩A) + P(A∩B) - 3P(A∩B∩C)


= 12.14+14.13+13.12−3(13.12.14)


= 14

Example 2:

If three coins are tossed together, what are the probability that the first shows head, the second shows tail and the third shows head?

Solution:

The probability that the first coin shows head = \[\frac {1} {2}\]


The probability that the second coin shows tail = \[\frac {1} {2}\]


The probability that the third coin shows head = \[\frac {1} {2}\]


P(A∩B∩C) = P(A).P(B).P(C) 


= 12.12.12


= 18

Example 3:

Solution:

(1−P(A))(1−P(B))(1−P(C))


Therefore, the required probability is:


= 1−[(1−12)(1−13)(1−14)]


= 1−[12.23.34]


= 1−14


= 34


Therefore, the probability that the sum gets solved is \[\frac {3} {4}\].

Example 4:

If a dice is rolled and a coin is tossed together, what is the probability that the coin gets a tail and the dice gets a 2?

Solution:

Probability of getting a tail =\[\frac {1} {2}\]


Probability of throwing a 2 = \[\frac {1} {6}\]


Therefore, getting a tail on a coin does not depend on getting a 2 on a dice and vice versa, the two events are independent. The probability that the two given events occur together is equal to the product of their respective probability.


Therefore, the required probability is: 12×16


= 112


Therefore, the probability that the coin shows tail and the dice shows 2 is \[\frac {1} {12}\].

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FAQs (Frequently Asked Questions)

1. What is the difference between Mutually Exclusive and Independent Events?

The following are some of the differences between the Independent event and Mutually Exclusive Event

Independent Events

Mutually Exclusive Events

The occurrence of the event cannot be specified based on the previous occurrence.

The occurrence can be predicted based on the previous occurrence.

It can have a common outcome.

It cannot have a common outcome.

If A and B are two independent events, then

P(A ∩ B) = P(B) .P(A)

If A and B are two mutually exclusive events, then

P(A ∩ B) = 0

2.  How are Independent Events related to Conditional Probability?

Conditional probability is the probability of the occurrence of one event in the case that a second event occurs. The conditional probability that an event A occurs, given that event B occurs is given by,


P\[(\frac {A} {B})\]\[(\frac {P(A∩B)} {P(B)})\]


However, if two events are independent, the occurrence of one event will not affect the occurrence of other. Hence the conditional probability of occurrence of event A, provided that event B will occur is equal to the probability of occurrence of event A, that is,


P\[(\frac {A} {B})\] = P(A) 

3. How to find the probability of an independent event?

For finding the probability of independent events one must know the formula of conditional probability which is given below: If the probability of events A and B is P(A) and P(B) respectively, then the conditional probability of event B is such that event A has already occurred is P(A/B). The conditional probability formula is presented below.


P(AB) = P(A∩B) P(B) or P(B∩A)P(B)


Given, P(A) must be greater than 0. P(A) less than 0 means that A is an impossible event. In P(A∩B), the intersection denotes the compound probability of an event.

4. What is an Independent event, explain with an example?

The event of an occurrence which does not depend on any other event is called an Independent event. For example, if a coin flipped in the air and got the outcome as Head, then again flipped the coin and got the outcome as Tail. In both cases, the occurrence of both events does not depend on each other. It is one of the events in probability. Here is the complete definition of independent events in probability.

5. Define Probability?

The meaning of probability is basically the extent to which something is likely to happen. Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event that is possible. 

6. What are the methods to identify independent events?

Step 1: Check if it is possible for the events to happen in any order? If yes, go to Step 2, or else go to Step 3


Step 2: Check if one event affects the outcome of the other event? If yes, go to step 4, or else go to Step 3


Step 3: The event is independent. Use the formula of independent events and get the answer.


Step 4: The event is dependent. Use the formula of dependent events and get the answer.

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