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The Relationship Between Conditional Probabilities & Independence

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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Conditional and independent probabilities are a basic part of learning statistics. It's important that you can understand the similarities and differences between the two as discussed in this lesson.

Conditional and Independent Probabilities

William is a horse enthusiast. He loves everything there is about horses and horse races. His favorite horse, Big Bertha, is about to race at the local horse racing track. Another horse is racing at the tracks, too. Sleepy Sally is a local favorite. William believes that Big Bertha will place first and Sleepy Sally will place second.

Meanwhile, William's friend Derek is watching his favorite basketball team play in the final game of their regional tournament. William promises Derek they will have a big celebration if Big Bertha wins first and Derek's team wins the tournament. William needs to understand the relationship between conditional probabilities and independent probabilities to understand the likelihood of these two scenarios.

Defining Conditional and Independent Probabilities

Before we understand William's probability problem, let's look at a common probability example to explain conditional probabilities and independent probabilities. Let's say you have a deck of cards. How many aces are in that deck of cards? Right, there are four aces in a deck of cards. What is the likelihood that you will pull two aces in a row from a deck of cards? This is an example of conditional probability.

Conditional probability is probability of a second event given a first event has already occurred. You can't find the probability of drawing two aces in a row if your first draw is a king. However, if your first draw is an ace, then you need to look at the deck in a whole new way to determine the probability of drawing a second ace.

You have two events happening in this scenario. To pull two aces from the deck, you must consider that pulling a second ace out of the deck is dependent upon you pulling out that first ace. When you pull the first ace from the deck, you are leaving only 3 aces and 51 cards left in the deck; therefore, this would be an example of conditional probability where the first event influences the probability of the second event. This is conditional probability with two dependent events.

A dependent event is when one event influences the outcome of another event in a probability scenario. Since pulling an ace from the deck changes the number of aces in the deck, this is an example of dependent events in a conditional probability.

What if you wanted to look at the probability of drawing just one ace from the deck? In this case, you would only have one event to consider. This is known as an independent event, which is when the probability of an event is not affected by a previous event.

You can also have conditional probability with two independent events. This happens when you have two events that can occur independently. For example, I might want to know the probability of pulling an ace out of a deck of cards while my friend pulls a green marble out of a bag of red and green marbles. The probability of you pulling an ace out of the deck won't influence the probability that your friend pulls a green marble out of the bag. These two events have nothing to do with one another, therefore they are independent events.

When we look at conditional probability, we write it like this in statistics: P(A|B) This is read, the probability of A given B, meaning what's the probability of A if event B also happens. P(B|A) is more common, and is read the probability of B given A, meaning what's the probability of B if event A also happens.

Conditional and Independent Probability Relationship

Let's get back to our scenario of William and Big Bertha. Remember that William wants to know the probability of Big Bertha getting first place and Sleepy Sally getting second place. We now know that is an example of conditional probability.

There are a lot of scenarios that could happen here. Big Bertha could get first place and another horse could get second place. Similarly, Sleepy Sally could get second place and another horse could get first place. Therefore, we could say that the probability of Sleepy Sally getting second place is dependent upon Big Bertha getting first place. Then these events would be dependent and not independent events.

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