# How to Apply Discrete Probability Concepts to Problem Solving

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• 0:05 Understanding Discrete…
• 1:35 Properties of Probability
• 4:18 Practice Problems
• 6:25 Lesson Summary
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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.

Discrete probability concepts, such as expected value, success, and failure, can be used to help you solve real-world problems and inform you when making decisions.

## Understanding Discrete Probability

Colin is starting his own business. He wants to sell ties that look like different objects. Right now, he has the banana tie, the sword tie, the fish tie, and the spoon tie. He wants to do some research to figure out which of the ties are the most popular and which ties he should no longer carry in his online store.

In this lesson, you will learn about discrete probability and the properties of probability. You'll also work through a practice problem to reinforce your learning. First, let's discuss discrete probability.

A discrete variable is an outcome of discrete data, which is data that cannot be divided; it is distinct and can only occur in certain values, meaning that a variable is a result of an experiment that cannot be divided. For example, if you were to count the number of people in a classroom, you would have a discrete variable because you can only have a whole person, not a half or a quarter of a person, in a classroom.

Discrete probability is the probability related to discrete data. In Colin's experiment, each tie is a discrete variable; his customers do not have a choice but to buy a full tie. The customers do not have an option to buy part of the tie. Right now, we can say that each tie has a 25% chance of being purchased when a customer buys something from Colin's online store.

We need to figure out more information before determining the most popular tie and other data for Colin's store. To do this, you'll need to understand more about the properties of probability.

## Properties of Probability

There are many properties of probability including the range, the success probability, the failure probability, and the expected value. In this section, we will briefly cover each of these concepts.

Probability ranges from 0 to 1, meaning you can have a 0% chance of something happening or a 100% chance of something happening. You can also have everything in between. However, you cannot have a probability that is less than 0, since a 0% chance means that something isn't going to happen anyway. And you can't have more than 100%, since 100% already means that you are guaranteed that something is going to happen.

Next, let's discuss the success probability and the failure probability. Each tie has a success probability and a failure probability. We already discussed earlier that each tie has an equal probability of being selected or a 25% chance. This is the success probability. In other words, this is the percent that tells us how likely a tie will be selected by an individual customer.

Now, the ties also have a failure probability, which is 1 - P. I want you to see that 1 in the formula as 100%. If a customer wants to buy a tie, then there is a 100% chance of a tie being purchased. If the banana tie has a 25% chance of being purchased by that customer, then what is the failure probability? 100 - 25 or 1 - .25. The banana tie has a 75% chance of not being selected or a 75% failure probability.

An expected value is the number of successful outcomes expected in an experiment. For example, we could find the expected value of out of 5 customers buying a tie, which is the expected value of a customer purchasing a spoon tie?

The formula for expected value of a discrete random variable is n * p. This is also considered the mean or average probability. The n represents the number of trials and the P represents the probability of success on an individual trial. Therefore, the expected value would be 5 * .25 = 1.25. In other words, out of 5 purchases, if each tie has equal popularity, you have a 100% (more, actually) chance of a spoon tie being purchased. Now, we know that the range for probability is between 0 and 1, so in this case, we can only say that there is a 100%, not 125% chance.

## Practice Problems

Colin has collected information about the sales of his ties over the past month. He has found that 60% of his customers purchase the banana tie, 20% of his customers purchase the sword tie, 15% of his customers purchase the spoon tie, and 5% of his customers purchase the fish tie.

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