Have you ever had a difficult time choosing options for a sandwich? Take a look at this dilemma.

For a sub sandwich, Luis has the following choices.

\begin{align*}& \underline{\text{Choice 1: Bread}} \qquad \underline{\text{Choice 2: Cheese}} \qquad \underline{\text{Choice 3: Meat}}\\ & \text{white, wheat} \qquad \quad \ \ \text{swiss, cheddar} \qquad \quad \ \text{turkey, ham, tuna}\end{align*}

How many different kinds of sandwiches can Luis make?

**One way of figuring this out is to use a tree diagram. By the end of the Concept, you will know how to help Luis with this dilemma.**

### Guidance

**Probability is a mathematical way of calculating how likely an event is likely to occur.**

**An** *event***is a result of an experiment or activity** that might include such things as:

- flipping a coin
- spinning a spinner
- rolling a number cube
- choosing an item from a jar or bag

An important concept when calculating probability is to think about outcomes.

**An** *outcome***is a possible result of some event occurring.** When you flip a coin, “heads” is one outcome; tails is a second outcome.

*Total outcomes***are computed simply by counting all possible outcomes.**

**That is a great question.**

One good way to count the total number of outcomes for an event is to make a ** tree diagram**.

**A tree diagram is a branching diagram that shows all possible outcomes for an event.**

**For instance, if you flip a coin two times, how many different outcomes are possible? To find out, make a tree diagram.**

To make a tree diagram, split the different events into either-or choices. The first choice breaks flip 1 down into heads or tails. Each outcome of flip 1 is broken down again for flip 2.

The pink box shows the total number of outcomes for both flips:

@$$\begin{align*}& \text{heads-heads} \quad \ \text{tails-heads}\\ & \text{heads-tails} \qquad \text{tails-tails}\end{align*}@$$

What happens when you increase the number of flips to three? Just add another section to your tree diagram.

**In all, there are now 8 total outcomes.**

@$$\begin{align*}& HHH \quad HTH \quad THH \quad TTH\\ & HHT \quad HTT \quad \ THT \quad TTT\end{align*}@$$

Use a tree diagram to calculate the total possible outcomes.

#### Example A

A car has four seats. How many different options are there if three people ride in the car?

**Solution: @$\begin{align*}12\end{align*}@$ options.**

#### Example B

Candice has two skirts, three shirts and two different sweaters. How many possible outfits can she create?

**Solution: @$\begin{align*}12\end{align*}@$ options.**

#### Example C

Sam is trying to create a pizza. He has two crust options, four different cheese options and 8 different toppings to choose from. How many pizzas can he make if he only chooses one topping?

**Solution: @$\begin{align*}64\end{align*}@$ options.**

Now let's go back to the dilemma from the beginning of the Concept.

**To figure this out, Luis can create a tree diagram to show all of his choices and calculate the sandwich outcomes.**

You can see that the tree diagram begins with the bread choices, then adds the second layer of the cheese options, and finally adds the meat choices.

**There are twelve possible sandwich outcomes for Luis.**

### Guided Practice

Here is one for you to try on your own.

How many possibilities are there for a win-win-win?

**Solution**

**You can see that when we break out all of the options that there is one chance for a win-win-win.**

### Video Review

### Explore More

Directions: Use a tree diagram to figure out all of the different outcomes.

- Jeff’s Jet Ski rentals has 3 different jet ski models: the single, the double, and the racer. Renters can rent for a half hour or a full hour. How many rental choices are there?
- CableCom offers Basic Cable, Premium Cable, and Super Premium Cable service. CableCom offers these services for home use, small business use, or large business use. How many different cable choices are there?
- The
*Gotham Gazette*offers the following newspaper choices:- home or office delivery
- weekdays only, weekends only, or all seven day delivery
- monthly or weekly payments

How many different kinds of choices can you get?

- On Main Street, Jiri has to go through 4 traffic lights that can be either red or green. How many different outcomes are there for the 4 lights?
- The I-Cone high tech ice cream shop offers the following options.
- cone: sugar, waffle
- size: teeny, mega, huge
- flavors: shocking blueberry, marvelous mango, chocolate attack

How many different choices are there?

- To remodel her kitchen, Gretchen has the following choices: Floor: tile or wood; Counter: Granite or formica; Sink: white, steel, stone. How many different choices can Gretchen make?
- Jeff has five different pairs of socks and three pairs of shoes. How many possible combinations are there?
- What if Jeff has six different pairs of socks and three pairs of shoes. How many possible combinations are there?
- What if Jeff has six different pairs of socks and four pairs of shoes. How many possible combinations are there?
- What if Jeff has eight different pairs of socks and three pairs of shoes. How many possible combinations are there?
- Jessie has three sweaters, two turtlenecks and three jackets. How many possible combinations are there?
- What if Jessie has two sweaters, three turtlenecks and three jackets. How many possible combinations are there?
- What if Jessie has four sweaters, three turtlenecks and three jackets. How many possible combinations are there?
- What if Jessie has three sweaters, two turtlenecks, two scarfs and three jackets. How many possible combinations are there?
- What if Jessie has four sweaters, two turtlenecks, two scarfs and three jackets. How many possible combinations are there?