Probability and Statistics > Probability > Dependent or Independent

## Dependent or Independent?

Watch the video or read the steps below:

Being able to tell the difference between a dependent and independent event is vitally important in **solving probability questions**. Why? Imagine a single event: winning the lotto. That **depends** upon you buying a ticket. So winning the lotto and buying a ticket are dependent events. Your odds of winning the lotto if you buy a ticket might be 1/1 million. But what about something unrelated, like driving to work and winning the lotto? Your odds of winning the lotto if you drive your car (and don’t buy a ticket) are zero. So the odds change **a lot** with different event types.

## What is an Independent Event?

An independent event is an event that has **no connection** to another event’s chances of happening (or not happening). In other words, the event has no effect on the probability of another event occurring. Independent events in probability are no different from independent events in real life. Where you work has no effect on what color car you drive. Buying a lottery ticket has no effect on having a child with blue eyes.

When two events are independent, one event does not influence the probability of another event.

**Simple examples of independent events:**

- Owning a dog and growing your own herb garden.
- Paying off your mortgage early and owning a Chevy Cavalier.
- Winning the lottery and running out of milk.

### What is a Dependent Event?

When two events are dependent events, one event influences the probability of another event.

**Simple examples of dependent events:**

- Robbing a bank and going to jail.
- Not paying your power bill on time and having your power cut off.
- Boarding a plane first and finding a good seat.

## How Can I Figure out what is a Dependent or Independent event?

Figuring out whether events are dependent or independent events can be challenging. Not all situations are as simple as they first appear. For example, you might think that your vote for president increases their chances of winning, but if you consider the Electoral College, that isn’t always the case.

You might think you have a chance of winning the top prize in a scratch off game. But the top prize might have already been won when you buy your ticket. For example, at the time of writing, if you purchased ten, two hundred “$1 Million Monopoly” scratch off tickets in Florida,

**your chances of winning are exactly the same: Zero!.**That’s because 0 out of 15 top prizes are remaining! States like Florida keep a “Remaining Prizes” list…but who really checks it?.

## Dependent or Independent? Steps

**Step 1:** *Ask yourself, is it possible for the events to happen in any order?* If no (the steps must be performed in a certain order), go to Step 3a. If yes (the steps can be performed in any order), go to Step 2. If you are unsure, go to Step 2.

Some examples of events that can clearly be performed in **any order** are:

- Tossing a coin, then rolling a die.
- Purchasing a car, then purchasing a coat.
- Drawing cards from a deck.

Some events that **must be performed in a certain order** are:

- Parking and getting a parking ticket (you can’t get a parking ticket without parking).
- Surveying a group of people, and finding out how many women are against gun rights (because you are splitting the survey into subgroups, and you can’t split a survey into subgroups without first performing the survey!).

**Step 2:** *Ask yourself, does one event in any way affect the outcome (or the odds) of the other event?* If yes, go to step 3a, if no, go to Step 3b.

Some examples of events that affect the odds or probability of the next event include:

- Choosing a card, not replacing it, then choosing another (because the odds of choosing the first card are 1/52, but if you do not replace it, your are changing the odds to 1/51 for the next draw).
- Choosing anything and not replacing it, then choosing another (i.e. choosing bingo balls, raffle tickets).

Some examples of events that do not affect the odds or probability of the next event occurring are:

- Choosing a card and replacing it, then choosing another card (because the odds of choosing the first card are 1/52, and the odds of choosing the second card are 1/52).
- Choosing anything, as long as you put the items back.

**Step 3a:** You’re done–the event is **dependent**.

**Step 3b:**You’re done–the event is **independent**.

That’s how to find out if an event is Dependent or Independent!

## Dependent or Independent Event Formulas in Probability

There are more formal ways to quantify dependent or independent events. You’ll come across these formulas in basic probability.

P(A|B) = P(A).

P(B|A) = P(B)

The probability of A, given that B has happened, is the same as the probability of A. Likewise, the probability of B, given that A has happened, is the same as the probability of B. This shouldn’t be a surprise, as one event doesn’t affect the other.

You can use the following equation to figure out probability for independent events:

**P(A∩B) = P(A) · P(B).**

**Example:**

A poll finds that 72% of Jacksonville consider themselves football fans. If you randomly pick two people from the population, what is the probability the first person is a football fan and the second is as well? That the first one is and the second one isn’t?

**Solution**: one person being a football fan doesn’t have an effect on whether the second randomly selected person is. Therefore, the events are independent and the probability can be found by multiplying the probabilities together:

First one and second are football fans: P(A∩B) = P(A) · P(B) = .72 * .72 = .5184.

First one is a football fan, the second one isn’t: P(A∩B) = P(A) · P(B) = .72 * 1 – 0.72) = 0.202.

In the second part, I multiplied by the complement. As the probability of being a fan is .72, then the probability of not being a fan is 1 – .72, or .28.

Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true. You can use the equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.

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