Independent and Dependent Events

What are Dependent Events vs Independent Events?

Definition

In mathematics, specifically statistics, events are often classified as dependent or independent. As a basic rule of thumb, the existence or absence of an event can provide clues about other events. Read on to find out more about dependent events vs independent events.

In general, an event is deemed dependent if it provides information about another event. An event is deemed independent if it offers no information about other events.

What are Dependent Events?

For events to be considered dependent, one must have an influence over how probable another is. In other words, a dependent event can only occur if another event occurs first.

While this is a mathematic/statistical term, speaking specifically to the subject of probabilities, the same is true of dependent events as they occur in the real world.

For example, say you’d like to go on vacation at the end of next month, but that depends on having enough money to cover the trip. You may be counting on a bonus, a commission, or an advance on your paycheck. It also most likely depends on you being given the last week of the month off to make the trip.

The primary focus when analyzing dependent events is probability. The occurrence of one event exerts an effect on the probability of another event. Consider the following examples:

1. Getting into a traffic accident is dependent upon driving or riding in a vehicle.
2. If you park your vehicle illegally, you’re more likely to get a parking ticket.
3. You must buy a lottery ticket to have a chance at winning; your odds of winning are increased if you buy more than one ticket.
4. Committing a serious crime – such as breaking into someone’s home – increases your odds of getting caught and going to jail.

What are Independent Events?

An event is deemed independent when it isn’t connected to another event, or its probability of happening, or conversely, of not happening. This is true of events in terms of probability, as well as in real life, which, as mentioned above, is true of dependent events as well.

For example, the color of your hair has absolutely no effect on where you work. The two events of “having black hair” and “working in Allentown” are completely independent of one another.

Independent events don’t influence one another or have any effect on how probable another event is.

Other examples of pairs of independent events include:

1. Taking an Uber ride and getting a free meal at your favorite restaurant
2. Winning a card game and running out of bread
3. Finding a dollar on the street and buying a lottery ticket; finding a dollar isn’t dictated by buying a lottery ticket, nor does buying the ticket increase your chances of finding a dollar
4. Growing the perfect tomato and owning a cat

Summary

In mathematics – namely statistics – as well as in real life, events are often categorized as either dependent or independent.

Dependent events influence the probability of other events – or their probability of occurring is affected by other events.

Independent events do not affect one another and do not increase or decrease the probability of another event happening.

“Dependent Events vs Independent Events”. 2022. Corporate Finance Institute. https://corporatefinanceinstitute.com/resources/knowledge/other/dependent-events-vs-independent-events/.

Dependent Events vs Independent Events

Aspect	Independent Events	Dependent Events
Definition	Events where the occurrence of one event does not affect the probability of the other.	Events where the occurrence of one event affects the probability of another.
Examples	Coin tosses Dice rolls Winning money at the casino and getting hit by a truck on the way home Picking balls out of jars/boxes with replacement Picking cards out of a deck with replacement	Picking balls out of jars/boxes without replacement Picking cards out of a deck without replacement The probability of getting the 3rd prize in a raffle after the first two prizes are given out
Why is this important?	If events are independent, then you can easily multiply events together to calculate probability.	With dependent events, you need to determine how the new probabilities are conditional to previous outcomes.
How does this affect the 3rd formula	No effect.	P(A + B) = P(A) * P(B) or P(A + B) = P(A) * PA(B) where P(A|B) and PA(B) means “the probability of B given that A has occurred”

Other tips to consider

• Watch for whether the question specifies “with” or “without” replacement when selecting objects.
• Dependent events can sometimes create two or more scenarios to consider.
• Whenever multiple events are said to be simultaneous, you can look at each event in turn. The word “replacement” with often determine the difference between independent and dependent probabilities, usually with balls or cards.

Umar, Bobby and Carl S. Pyrdum. Barron’s GMAT. Hauppauge, NY: Barron’s Educational Series, Inc., 2014.

Additional References

“Dependent and Independent Events – Probability – GeeksForGeeks”. 2021. GeeksForGeeks. https://www.geeksforgeeks.org/dependent-and-independent-events-probability/.

“Independent And Dependent Events: Probability”. 2022. theproblemsite.com. https://www.theproblemsite.com/reference/mathematics/probability/independent-and-dependent-events.

“Independent Events In Probability (Definition, Venn Diagram & Example)”. 2022. BYJUS. https://byjus.com/maths/independent-events/.

Videos

In this video, you will learn the difference between independent events, and dependent events. At the same time, you will learn how to calculate the probabilities of each.

This probability video tutorial provides a basic introduction into independent and dependent events. It provides example problems using colored marbles.