Video: Representing Sample Spaces with Tree Diagrams

A pizza parlor offers customizable pizzas. Customers can choose one of the two options for the base, cheese, and sauce and then add an optional topping. Using tree diagrams or otherwise, represent all possible pizza combinations and determine how many different pizzas there are.

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Video Transcript

A pizza parlor offers customizable pizzas. Customers can choose one of the two options for the base, cheese, and sauce and then add an optional topping. Using tree diagrams or otherwise, represent all possible pizza combinations and determine how many different pizzas there are.

This question is asking us to do two different things. First, represent all possible pizza combinations. And second, determine how many different pizza combinations there are. Let’s use a tree diagram to represent all possible pizza combinations.

We’ll start our tree diagram with the first base option, a thin crust. So let’s say we chose a thin crust. From there, you have two options for your cheese: cheddar or mozzarella. If you chose cheddar, you would then have to choose from your two sauces: tomato or barbecue. If you chose mozzarella, you would still have the choices between tomato and barbecue. And then you get to your choice of toppings.

Once you’ve chosen tomato sauce, you then have toppings that you can choose from: pepperoni, pineapple, or mushroom. At this point, we need to stop and think very carefully. The fourth category, the toppings are optional. And this means we need a fourth option for none. What if you’re ordering a pizza and you want no extra toppings?

This last stage is four choices instead of three: pepperoni, pineapple, mushrooms, or none. If you choose barbecue sauce, you’ll have those same four topping choices: pepperoni, pineapple, mushroom, or none. If you chose thin crust, mozzarella, tomato sauce, you have four topping choices: pepperoni, pineapple, mushroom, or none. And if you chose thin crust, mozzarella, barbecue sauce, you would have four topping choices: pepperoni, pineapple, mushroom, none.

And this entire tree graph is only half of the choices. This only represents all of the pizzas made with a thin crust. So we need to repeat this tree diagram for a deep pan pizza. Deep pan will have the same amount of choices: cheddar or mozzarella, tomato or barbecue sauce, and then the choices of pepperoni, pineapple, mushroom, or no toppings.

The second question asks us to determine how many different pizzas there are. Each of our smallest branches represent a pizza that’s customizable. For example, this first branch is a thin crust pizza with cheddar cheese, tomato sauce, and pepperonis.

In order to count the total number of different pizzas, we need to total these last branches. Each end has four pieces. So we can total them in this way. There are four plus four plus four plus four pizza options that have a thin crust. And there are four plus four plus four plus four options with a deep pan: 16 thin crust options, 16 deep dish options. 16 plus 16 represents the 32 total number of different pizza combinations there are. Given these options, you could choose from 32 different pizzas.

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