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Applying Conditional Probability & Independence to Real Life Situations

Applying Conditional Probability & Independence to Real Life Situations
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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.

It can be really confusing learning how to apply conditional and independent probability to real-life situations. This lesson focuses on several examples and practice problems to help you learn how to find conditional probability.

Probabilities in Real-Life Scenarios

Andrew is getting ready for a business conference. His plane leaves in a few hours, so he will have to hurry. He grabs a tie out of his closet, two socks out of his dresser, and two shoes from under his bed. Andrew begins to dress and notices that there is something wrong with this scenario - some of the items he chose do not match his suit. Andrew can understand more about this scenario if he learns about conditional probability and independent probability.

In this lesson, you will learn about Andrew's dilemma and how it applies to conditional and independent probabilities. You will also learn how to find both conditional and independent probabilities.

Defining Conditional and Independent Probabilities

When Andrew grabs a tie out of his closet without looking, this is an example of independent probability. In this case, you only have one event to consider. Independent probability is when the probability of an event is not affected by a previous event.

Andrew has 12 ties in his closet. Out of the 12, Andrew has four ties that match his suit. Andrew is only pulling one tie from the closet. That is one event that has no relation to any of the other events, which is what makes it an independent probability.

Andrew has many loose socks in his dresser drawer. He has 15 socks that are black with small stripes and 15 socks that are plain black. Andrew has to pull out one sock; then the second sock Andrew pulls out must match the first sock. Andrew having two matching socks is dependent upon which sock he pulls out first and which sock he pulls out second. In this case, you have two events to consider. This is an example of conditional probability, which is probability of a second event happening given that a first event has already occurred. This particular case of conditional probability deals with dependent events, which is when one event influences the outcome of another event in a probability scenario.

Andrew has six pairs of tennis shoes, two pairs of sandals, and one pair of dress shoes. When Andrew grabs a shoebox out from under his bed, this is a single event. Therefore, the probability of Andrew grabbing dress shoes out from under his bed is an independent event. However, if we want to know the probability of Andrew pulling a pair of dress shoes out from under his bed given that he has already pulled a tie that matches his suit from his closet, this would be a case of conditional probability with independent events.

Finding Conditional and Independent Probabilities

We can find the probability of Andrew getting the correct tie by finding the number of desired outcomes divided by the number of total possible outcomes. Again, this is a case of independent probability. We can use this formula:

P(A) = A/T

In this formula, P(A) represents the probability of event A, which in this case is the tie that matches the suit. A represents the number of ties that match Andrew's suit, and T represents the total number of ties. You will often see P(A) and A in probability problems. You won't see T in a formula very often, and we are just using it today as a placeholder.

Remember, Andrew has four ties that match his suit and a total of 12 ties in his closet. Therefore, the four ties that match Andrew's suit are the desired outcomes, and the 12 ties are the total possible outcomes. Our formula will look like this:

P(A) = 4/12

To simplify, Andrew has a 1 in 3 chance that he will get a tie that matches his suit.

Now we need to find the probability that Andrew will pull two matching socks from his drawer. The formula for conditional probability with dependent events is slightly different than independent probability.

The bar between A and B means 'given.' Therefore, the beginning of this problem reads 'the probability of B given A.' In other words, what is the probability of event B happening if A has already occurred?

Let's break this down one step at a time. First, what are the total number of socks in the drawer? Right; Andrew has 30 socks total in his drawer. That means there are 30 total possible outcomes. Now, Andrew has a black suit, so he wants his socks to match his suit. How many black socks are in Andrew's drawer? 15. Correct; there are 15 black socks. The first part of our formula will look like the independent probability formula:

15/30 P(A)

But what about the second sock? Remember, the second event is dependent on the first event actually happening. So, let's say that Andrew pulled a black sock from his drawer. Now how many socks are left in the drawer? 29 socks are left in the drawer because Andrew has already pulled one sock out. We are assuming that he is pulling out a black sock; so now, how many black socks are left in the drawer? 14 black socks are left in the drawer. Therefore, the conditional probability of the dependent events would be:

14/29 or P(B|A)

Now that you know how to find conditional probability with dependent events, let's look at conditional probability with independent events. Andrew now must pull his shoes out from under the bed. Andrew has six pairs of tennis shoes, two pairs of sandals, and one pair of dress shoes. The probability of Andrew pulling his dress shoes out from under the bed is one out of nine chances. That is because there is one pair of dress shoes and nine pairs of shoes total. We can notate that like this:

1/9 = P(B)

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