What is Independent and Dependent Events: Introduction
Probability theory is an essential topic for students studying mathematics at a higher level. For example, the weather forecast in some locations predicts that it will rain 50% of the time today.
The probability is the likelihood of an event occurring. The term "event" refers to one or more outcomes. The term "event" refers to a possible consequence. Total events are all possible outcomes related to the experiment posed in the question. Favorable events are sometimes known as events of interest.
This article will cover every topic in detail and at the end of the article one will be able to answer what is independent and dependent events, and their differences etc.
What is Independent and Dependent Events?
The probability or chance of an event occurring is intuitively believed to be the probability of it occurring. In the most basic circumstances, the probability of a specific event A occurring as a result of an experiment is calculated by dividing the number of ways A can occur by the total number of possible outcomes.
Independent Event:
An independent event has no bearing on the likelihood of another event occurring (or not occurring). In other words, the occurrence has no bearing on the likelihood of a subsequent event occurring.
Independent occurrences in probability are analogous to independent events in reality. What color car you drive has nothing to do with where you work. Purchasing a lottery ticket does not affect the likelihood of having a child with blue eyes.
When two events are independent, one event does not affect the likelihood of the other.
Dependent Event:
When two events are dependent on each other, one event influences the likelihood of the other. A dependent event is dependent on another event to occur first.
Dependent events in probability are analogous to dependent occurrences in reality: If you want to go to a performance, it may be contingent on whether you receive overtime at work; if you want to see family out of the country next month, it may be contingent on whether you can obtain a passport in time.
In more technical terms, when two occurrences are dependent, the occurrence of one influences the likelihood of the other.
Example of Independent and Dependent Events:
The following are examples of independent events.
Purchasing a lottery ticket and discovering a penny on the floor (your chances of discovering a penny are not affected by purchasing a lottery ticket).
Taking a cab home and watching your favorite movie on television.
Getting a parking ticket and going to the casino to play craps.
The following are examples of dependent events.
Getting a parking citation for parking illegally. Parking illegally raises your chances of receiving a citation.
Purchasing ten lottery tickets and winning. The more tickets you purchase, the better your chances of winning.
Getting into a car accident while driving.
Differentiate Between Independent and Dependent Events:
Independent and Dependent Events Formulas:
There are more formal methods for determining whether an event is dependent or independent. These formulas can be found in basic probability.
P(A|B) = P(A).
P(B|A) = P(B)
Given that B has occurred, the chance of A is the same as the probability of A. Similarly, provided that A has occurred, the chance of B is the same as the probability of B. This is hardly surprising given that one occurrence does not affect the other.
To calculate the probability of independent events, use the following equation:
P(A∩B) = P(A) · P(B)
Summary
Events can be classified as independent or dependent in mathematics and real life. Dependent events are influenced by or affect the probability of other events, while independent events do not have any effect on each other's likelihood of occurring. When the outcome of one event determines the outcome of another, they are considered dependent, and the probability of both events occurring is the product of their individual probabilities. On the other hand, independent events occur without altering the likelihood of each other, and the probability of both events occurring is equal to the product of their individual probabilities.
FAQs on Difference Between Independent and Dependent Events for JEE Main 2024
1. What is independent and dependent events? Give examples.
An independent event has no bearing on the likelihood of another event occurring (or not occurring). Independent occurrences in probability are analogous to independent events in reality.
For example,
Having a dog and maintaining your herb garden.
Owning a Chevy Cavalier and paying off your mortgage early.
The combination of winning the lotto and running out of milk.
A dependent event is dependent on another event to occur first. Dependent events in probability are analogous to dependent occurrences in reality: If you want to go to a performance, it may rely on whether you receive overtime at work; if you want to see family out of the country next month, it may depend on whether you get overtime at work.
For example,
Going to jail for robbing a bank.
Failure to pay your electricity bill on time may result in your power being turned off.
Getting on a plane and finding a decent seat first.
2. Differentiate between independent and dependent events.
The primary difference between independent and dependent events is that one of them gets influenced by the results of previously occurring events. Dependent events are two or more occurrences that are dependent on one another. If one event is altered by chance, another is likely to differ.
Independent events are ones whose occurrence is unrelated to any other event. If the chance of an event A occurring is not altered by the occurrence of another event B, then A and B are said to be independent events.
3. How to check whether the event is independent?
Remember that two events are independent if neither affects the other. That is, knowing that one event has already occurred has no bearing on the likelihood that another event will occur. How can we use probabilities to determine whether two events are independent? There are three straightforward methods for determining independence:
Is P(A) × P(B) = P(A and B)?
Is P(B|A) = P(B)?
Is P(A|B) = P(A)?
If you respond yes to any of these three questions, then occurrences A and B are distinct. Furthermore, if any of these three is true, the others are also true; hence, you only need to prove that one of them is true.
4. How to demonstrate that the two events are dependent?
Two events A and B are said to be independent if the occurrence of one does not alter the likelihood of the occurrence of the other. If the occurrence of one event affects the likelihood of the occurrence of the other, the two events are said to be dependent.
5. What is an event in probability?
An event is an outcome of an experiment in probability, or it is defined as a series of outcomes of an experiment to which a probability is assigned. In probability, a dependent event affects the outcome of another event, whereas an independent event does not affect the outcome of another event.