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.

Colin hosts an event to celebrate the opening of his first tie storefront. When the store opens, 32 customers come rushing in. Can you find the expected value of how many customers will purchase the banana tie? What is the success probability of the banana tie? What is the failure probability of the banana tie? Pause the video here to find the answer.

How did you do? The probability of success for the banana tie is 60% and the probability of failure is 40%. The expected value for this particular scenario is 19.2, meaning that approximately 19 people will leave the store with a banana tie.

Colin has decided to discontinue his fish tie. After a month of more data collecting, he has found that 60% of customers purchase his banana tie and 35% of customers purchase the spoon tie. Colin does a check up on his new store. There are 64 customers. Assuming that all of the customers are going to purchase a tie, what is the expected value of the sword tie? What is the probability of success and failure? Pause the video here to find your answer.

What’s your answer? You have to do a little more calculating here to find the success and failure probabilities of the sword tie. Since we know the success probabilities of the banana tie is 60% and the spoon tie is 35%, we can add these together and subtract from 100% to get the success probability of the sword tie. The probability of success is 5% while the probability of failure is 95%. The expected value is 3.2. This means that out of the 64 people that purchase a tie, it’s only probable that three of those customers will leave with a sword tie.

## Lesson Summary

Remember, today we worked with a discrete variable, which is an outcome of **discrete data**, or data that cannot be divided; it is distinct and can only occur in certain values. For example, if one of Colin’s customers picked out one particular tie, it would be a discrete variable because they can’t pick out only half of a tie! Today, we found **discrete probability**, which is the probability related to discrete data.

We also discussed some properties of probability. Remember, 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.

We discussed the success probability and the failure probability. You are usually given the success probability. To find the failure probability, simply use the formula 1 – *P*.

An **expected value** is the number of successful outcomes expected in an experiment. The formula for expected value of a discrete random variable is *n* * *p*.

## Learning Outcomes

Review this lesson several times, then subsequently verify that you can:

- Write the definition of discrete probability
- Identify the various properties of probability
- Use the formula to solve for discrete probability