The Multiplication Rule of Probability is a concept you will use frequently when solving probability equations.

In this lesson, learn the two different scenarios in which you will use the multiplication rule of probability.

## The Multiplication Rule

Steve is campaigning to be a county commissioner on the board for his county. His friend, Gary, is a professor at the local college and has agreed to help Steve with his campaign. Gary and Steve are putting together campaign buttons for a rally they are both hosting in a few days.

There are 30 buttons total, 13 buttons are blue and 17 buttons are red. Gary puts all of the buttons into a bag. Steve and Gary both want to wear red buttons to the rally.

What is the probability that Gary will pull two red buttons in a row out of the bag without looking? To solve this problem, you need to understand the Multiplication Rule of Probability. The **Multiplication Rule of Probability** means to find the probability of the intersection of two events, multiply the two probabilities.When you want to know the probability of two events occurring, that is called the intersection of the two events. The Multiplication Rule of Probability is used to find the intersection of two different sets of events, called independent and dependent events.

**Independent events** are when the probability of an event is not affected by a previous event. A **dependent event** is when one event influences the outcome of another event in a probability scenario. To find the intersection of two events, whether they are independent or dependent, multiply the two probabilities together.Steve and Gary’s button scenario is an example of dependent events. This is a slightly more complicated problem, so let’s start with independent events first.

## Multiplying Independent Events

Gary wants to help Steven in researching the voting trends in each of the towns in the county. Steve covers the two towns pretty well in his campaign.

Steve talks to the townspeople about his policies. He passes out flyers and puts up signs in the two towns to spread the word about his candidacy. After doing this, Gary and Steve collect data about each town’s feelings towards Steve as a candidate.

Steve wants to know the probability of winning over two towns in the county. This is an example of independent events, because we are assuming that one town’s voting patterns do not affect or depend on the other town.In Town A, one out of every four people felt favorable towards Steve as a candidate.

Steve has a 25% chance of winning the votes from Town A.In Town B, five out of every seven people felt favorable towards Steve as a candidate. Steve has a 71% chance of winning the votes from Town B.Steve asks Gary, what is the probability of winning the votes in both towns? To figure this out, Gary will use the multiplication rule of probability and this formula: P(A and B) = P(A) * P(B).This is the multiplication rule for two independent events. This formula is read: the intersection of event A and event B equals event A multiplied by event B.

Event A is the probability of town A voting for Steve, and Event B is the probability of town B voting for Steve. To find the probability of both of these events happening, you can use the Multiplication Rule of Probability by simply multiplying the two probabilities like this: 1/4 * 5/7 = 5/28 or approximately 18%.To find the probability of two independent events occurring at the same time, simply multiply the two probabilities together. Remember, this is the intersection of two independent events.

This is not looking good for Steve! Hopefully, he will have better luck in the other towns.

## Multiplying Dependent Events

Let’s return to our opening scenario. There are 30 buttons total, 13 buttons are blue and 17 buttons are red. What is the probability that Gary will pull two red buttons in a row out of the bag without looking? The first time Gary pulls a button out he has the probability of 17/30, or approximately 57%.

The numerator in this probability exercise is 17, the total number of red buttons. The denominator in this probability exercise is 30, the total number of buttons.If he does not put this button back in to the bag, then the second probability of Gary pulling a red button from the bag is 16/29, or approximately 55%.

The numerator in this probability exercise is 16, the total number of red buttons remaining in the bag. The denominator in this probability exercise is 29, the total number of buttons remaining. To find the intersection of these two events using the multiplication rule of probability, simply multiply the two probabilities together: 17/30 * 16/29 = .

308 or approximately 31%.This means that Gary has a pretty good chance of pulling out two red buttons independently, but for both to happen together is pretty unlikely!

## Practice Problems

Now that you understand the multiplication rule of probability, let’s practice with a few examples.

#### Example 1

Lisa is baking cookies. She bakes a batch of a dozen cookies with peanut butter and chocolate chips and one batch of a dozen cookies with just chocolate chips. When she finishes, she places both batches on to a platter together. What is the probability of someone selecting two plain chocolate chip cookies in a row?The correct answer is approximately 24%.

The second cookie selection is dependent upon the first cookie selection, since the person will not be putting the cookies back. Therefore, the first cookie selection changes the probability of the second cookie selection.The first probability will be 12/24, or 50%. The first number is the number of plain cookies, and the second number is the total number of cookies.The second probability is 11/23. The first number is the number of plain cookies, minus the first one that was selected. The second number is the total number of cookies, minus the first cookie that was selected.

Now multiply the two probabilities to find the intersection using this formula: 12/24 * 11/23 = 24%.

#### Example 2

Lisa, Gary and Steve are playing a game at their weekly game night get-together. They decide to play Pretzel first, a game that has different colored squares on a mat where each player places a hand or a foot on a different color depending on the spinner. There are two different spinners, one is labeled right hand, left hand, right foot and left foot.

The second spinner is labeled with colors; there are four purple squares, three blue squares, four pink squares and three orange squares randomly arranged on the spinner. What is the probability of Gary getting left hand, purple square?The correct answer is seven percent. The two spinners are separate spinners, and they do not effect one another. Therefore, we have two independent events. We can determine the probability of the intersection of these independent events by using this formula: P(A and B) = P(A) * P(B).The probability of getting left hand is one out of four, since there are four selections on the spinner and only one is the left hand.

The probability of getting purple square on the second spinner is four out of 14, since there are four purple squares with a total of 14 squares on the spinner. To find the intersection of these independent events, simply multiply the two events like this: 1/4 * 4/14 = .07 or 7%.

## Lesson Summary

You will often have to find the intersection of two events in probability. The intersection of the two events can be found by using the **Multiplication Rule of Probability**, which means to find the probability of the intersection of two events, multiply the two probabilities.

How you use this rule is dependent upon the type of events you are working with, independent or dependent. **Independent events** are when the probability of an event is not affected by a previous event. A **dependent event** is when one event influences the outcome of another event in a probability scenario. You can use the multiplication rule with many probability exercises, just make sure you understand the types of events you are working with first!

## Learning Outcomes

After you are done with this lesson you should be able to:

- State the multiplication rule of probability
- Calculate the probability of independent and dependent events