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Counting Events

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Counting Events

Have you ever been in a talent show?

Alicia is going to sing for the Talent Show. She is very excited and has selected a wonderful song to sing. She has been practicing with her singing teacher for weeks and is feeling very confident about her ability to do a wonderful job.

Her performance outfit is another matter. Alicia has selected a few different skirts and a few different shirts and shoes to wear. Here are her options for shirts

Striped shirt

Solid shirt

Here are her options for skirts.

Blue skirt

Red skirt

Brown skirt

Here are her options for shoes

Dance shoes

Black dress shoes

How many different outfits can Alicia create given these options?

This is best done using a tree diagram. Alicia needs to organize her clothing options using a tree diagram. This Concept will show you all about tree diagrams. When finished, you will know how many possible outfits Alicia can create.

Guidance

Nadia’s soccer team has 2 games to play this weekend. How many outcomes are there for Nadia’s team?

A good way to find the total number of outcomes for events is to make a tree diagram. A tree diagram is a branching diagram that shows all possible outcomes for an event.

To make a tree diagram, split the different events into either-or choices. You can list the choices in any order. Here is a tree diagram for game 1 and game 2.

As you can see, there are four different outcomes for the two games:

& \text{win-win} && \text{win-lose}\\& \text{lose-win} && \text{lose-lose}

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

In all, there are 8 total outcomes.

A tree diagram is a great way to visually see all of the options possible. It can also help you to organize your ideas so that you don’t miss any possibilities.

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?

First, let’s create a tree diagram that shows all of the possible options.

Step 1 : List the choices.

Choice 1 Floor : tile, wood Choice 2 Counter : granite, formica Choice 3 Sink : white, steel, stone

Step 2 : Start the tree diagram by listing any of the choices for Choice 1. Then have Choice 1 branch off to Choice 2. Make sure Choice 2 repeats for each branch of Choice 1.

Step 3 : Fill in the third choice. We have left some of the spaces for you to fill in. Double click to check your answers.

Step 4 : Fill in the outcomes. Again some of the spaces are left for you to fill in. Double click to check your answers.

You can see that there are 12 possible outcomes for the kitchen design.

Calculate the number of possible outcomes in each example. You can draw a tree diagram to help you.

Example A

Jeff has five different pairs of socks and three pairs of shoes. How many possible combinations are there?

Solution: 15 combinations

Example B

Jessie has three sweaters, two turtlenecks and three jackets. How many possible combinations are there?

Solution: 18 combinations

Example C

Kelly has chocolate or vanilla ice cream, three choices of toppings and four sauces.

Solution: 24 combinations

Here is the original problem once again. Reread it and then look at the tree diagram created.

Alicia is going to sing for the Talent Show. She is very excited and has selected a wonderful song to sing. She has been practicing with her singing teacher for weeks and is feeling very confident about her ability to do a wonderful job.

Her performance outfit is another matter. Alicia has selected a few different skirts and a few different shirts and shoes to wear. Here are her options for shirts

Striped shirt

Solid shirt

Here are her options for skirts.

Blue skirt

Red skirt

Brown skirt

Here are her options for shoes

Dance shoes

Black dress shoes

How many different outfits can Alicia create given these options?

This is best done using a tree diagram. Alicia needs to organize her clothing options using a tree diagram. To do this, we can take each option and create a diagram to show all of the options.

Based on this tree diagram, you can see that Alicia has twelve possible outfits to choose from.

Vocabulary

Tree Diagram
a visual way of showing all of the possible outcomes of an experiment. Called a tree diagram because each option is drawn as a branch of a tree.

Guided Practice

Here is one for you to try on your own.

A local pizza place offers a pizza with several options. On a basic pizza, you can choose one of each option. The options are a sourdough or whole wheat crust, two types of cheese and eight toppings.

Based on these possibilities, how many possible outcomes are there?

Answer

To figure this out, you can draw a tree diagram or you can simply multiply.

Drawing a tree diagram might look nice, but it can take a lot of time.

To save time, let's multiply.

Our choices are two crusts, two cheeses and eight toppings.

2 \times 2 \times 8 = 32

There are 32 different pizza options.

Video Review

This is a Khan Academy video on probability and outcomes.

Practice

Directions: Use Tree Diagrams for each of the following problems.

1. The Triplex Theater has 3 different movies tonight: Bucket of Fun, Bozo the Great, and Pickle Man. Each movie has an early and late show. How many different movie choices are there?

2. Raccoon Stadium offers the following seating plans for football games:

  • lower deck, middle loge, or upper bleachers
  • center, side, end-zone

How many different kinds of seats can you buy?

3. Cell-Gel cell phone company offers the following choices:

  • Free internet plan or Pay internet plan
  • 1200, 2000, or 3000 minutes
  • Premium or standard phone

How many different kinds of plans can you get?

4. Jen’s soccer team is playing 4 games next week. How many different outcomes are there for the four games?

5. The e-Box laptop computer offers the following options.

  • Screen: small, medium, or large
  • Memory: standard 1 GB, extra 2 GB
  • Colors: pearl, blue, black

How many different combinations are there for options?

6. Jesse has four pairs of pants and three shirts. How many different outfits can he make?

7. If Jesse adds three sweaters, how many different outfits can he make now?

8. Kyle has selected three different paint colors and two trim colors for his bedroom. How many different color combinations can he create?

9. There are four teams playing in a tournament. Each team will play two sports. How many different combinations are there for teams and sports?

10. Candice can choose between the following electives.

  • German
  • Gymnastics
  • Chorus
  • French

If she can choose two electives, how many different options are there of choices?

11. If we add Track to her options, how many two elective options are there?

12. Miles is making a salad. He wants to put three things on it. Here are his choices.

Tomatoes

Pickles

Onions

Cucumbers

Peppers

How many different salads can he create given these options?

13. If Miles adds cheese, how many different salads can he create given these options?

14. Muffins can have two additions to them. Here are the choices: blueberries, raspberries, nuts, cherries, strawberries. How many muffin combinations can there be?

15. If we add apples and cranberries, how many combinations can there be now?

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