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%.
