# Conditional Probability: Definition & Uses

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• 1:08 Dependent Events
• 2:58 Notation for…
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Lesson Transcript
Instructor: Joseph Vigil
We know a coin can land on either heads or tails. But what would happen if one coin flip changed the next? In this lesson, we'll look at events that are dependent on each other, and we'll learn how to calculate the probability of two events occurring in a combined manner.

## Probability

What are the chances your favorite baseball team will make it to their league's playoffs? What are the chances that they make the World Series? Or even that they'll win the World Series? The answers to all these questions depend on probability, which is a numerical expression of how likely it is that an event will occur.

Let's flip a coin, for example. It landing will be one of two sample points, or possible outcomes: heads or tails. These two sample points make up our sample space, the complete set of possibilities in an experiment. Since one out of the two sample points is heads, the probability of the coin landing on heads is a 1 out of 2, or 1/2. Likewise, since one out of the two sample points is tails, the probability of the coin landing on tails is 1 out of 2, or 1/2.

That's simple enough. Every time we flip the coin, we have the same sample space. Even if we flipped it a million times, the next flip will still give either heads or tails. But what if the sample space changed every time we ran an experiment?

## Dependent Events

You've got to get to work! You've slept past your alarm, and it's still dark. You need two black dress socks to finish dressing. You have two white socks and two black ones in your drawer. What are your chances of grabbing the right socks?

Let's look at the socks we have now and how that group of socks may change. When you grab your first sock, it can of course be black or white. Let's say you get lucky, and you grab a black sock on your first try. Since 2 out of the 4 socks are white, you have a 2/4, or 1/2, chance of picking a black sock.

Unlike the coin flip, though, the odds of grabbing a certain sock change every time we run this experiment. This is an example of conditional probability, which is the probability of one event happening given that another event has already happened. In other words, the second event is conditional based on the previous event.

Now you have one black sock and two white socks left. The group of socks has changed. So the odds of picking either a white or black sock has also changed. Each time you pick a sock, you change future options. In other words, each selection is dependent on the one before it; so each selection is a dependent event.

Now that there are two white socks and one black sock left, you have a 1/3 chance of picking a black sock. Since you have a 1/2 chance of picking a black sock the first time and a 1/3 chance of picking a black sock the second time, we'll multiply those two probabilities together. 1/2 * 1/3 = 1/6. You have a one out of six chance of picking two black socks right away. Not very good odds. We'd better turn on the lights!

## Notation for Conditional Probability

We have to use a common language so people can be on the same page when talking about probabilities. So, when we write P(A), we mean the probability that some event, A, will happen.

Let's go back to our coin. If A represents the coin landing on heads, then P(A) = 1/2, because there's a 1 out of 2 chance that the coin will land on heads.

If B represents the coin landing on tails, then P(B) also equals 1/2, because there's a 1 out of 2 chance that the coin will land on tails.

When we were dealing with the socks, though, each event was dependent on those before it. So, we write P(B|A), which means the probability that B happens if A has already happened, also known as the conditional probability of B given A.

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