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# 11.1: Counting Methods

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Bike Shop

Telly and Carey are working in a bicycle shop over the spring vacation. They are excited because Ms. Kelley the owner is going to let each of them choose a new bike when the week is done. This way, they get to work and earn a new bike at the same time.

There are so many different types of bikes, that Telly is having a difficult time choosing. To complicate matters even more, Ms. Kelley said that she would let Telly design her own bike so she can choose seat color and type of handlebars as well as color of the bike. Telly knows that she wants a mountain bike, so at least that part has been chosen. Here are all of the options that Telly has.

Mountain bike

Colors = Red, Green, Blue or Purple

Seat = normal or extra cushion

Handlebars = straight or curved

Telly took out a piece of paper and tried drawing out all of her options. She became frustrated almost immediately.

This is where you come in. You can help Telly to work through this problem by learning about the Counting Principle and about tree diagrams. Pay attention because you will see this problem again.

What You Will Learn

In this lesson, you will learn how to use the following skills.

• Use tree diagrams to list all possible outcomes of a series of events involving two or more choices or results.
• Recognize the number of possible outcomes of a series of events as the product of the number of possible outcomes for each event.
• Use the Counting Principle to find all possible outcomes of a series of events involving two or more choices or results.
• Find probabilities of specified outcomes using the Counting Principle.

Teaching Time

I. Use Tree Diagrams to List All Possible Outcomes of a Series of Events Involving Two or More Choices or Results

In this chapter you will learn about probability. 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. For example, 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 example, 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:

heads-heads tails-headsheads-tailstails-tails\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.

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

Example

For a sub sandwich, Luis has the following choices.

Choice 1: BreadChoice 2: CheeseChoice 3: Meatwhite, wheat  swiss, cheddar turkey, ham, tuna\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?

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.

II. Recognize the Number of Possible Outcomes of a Series of Events as the Product of the Number of Possible Outcomes for Each Event

Sometimes it just isn’t possible to draw out a whole tree diagram to calculate outcomes. When this happens, we can use arithmetic to calculate the outcomes. Let’s look at an example.

Example

Ichiro’s Car Wash offers three different wash services – basic outer body wash, interior cleaning, and finally, custom hand-detailing. Customers order a wash choice plus wax or no wax. You can use a tree diagram to find that there are 6 different choices, or outcomes, for a car wash.

How could we do this without a tree diagram?

We can look at the situation in terms of outcomes. For the first choice there are 3 different outcomes. For the second choice there are 2 different outcomes.

3 outcomes2 outcomes=6 outcomes\begin{align*}3 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 6 \ \text{outcomes}\end{align*}

What if we changed the options? Would this method still work? Let’s add on to the car wash problem and see.

Consider Ichiro’s oil change choices. Customers can get standard or synthetic oil, filter or no filter, and pay with a coupon or for the regular price. For each choice there are 2 different outcomes.

2 outcomes2 outcomes2 outcomes=8 outcomes\begin{align*}2 \ \text{outcomes} \cdot 2 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 8 \ \text{outcomes}\end{align*}

This method of calculating the number of total outcomes can be stated as a general rule called the Counting Principle.

Counting Principle: The number of choices or outcomes for two independent events, A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} taken together, is the product of the total number of outcomes for each event.

Total outcomes for A and B=(number of outcomes for A)(number of outcomes for B)\begin{align*} \text{Total outcomes for} \ A \ \text{and} \ B = \text{(number of outcomes for} \ A) \cdot \text{(number of outcomes for} \ B)\end{align*}

Now that you understand that we need to calculate choices or outcomes to calculate all of the possible number of outcomes, let’s look at applying the Counting Principle in some examples.

III. Use the Counting Principle to Find All Possible Outcomes of a Series of Events Involving Two or More Choices or Results

Once again, the Counting Principle requires that we take the number of choices or outcomes for two independent events and multiply them together. The product of these outcomes will give us the total number of outcomes for each event.

For example, 2 tosses of a coin there are 2 outcomes for each toss. Using the Counting Principle, you can find the total number of outcomes as:

2 outcomes2 outcomes=4 total outcomes\begin{align*}2 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 4 \ \text{total outcomes}\end{align*}

You can use the Counting Principle to find probabilities of events. For example, suppose you wanted to know the probability of rolling two number cubes and coming up with a sum of 5. The probability of any event is equal to the ratio of favorable outcomes to the total number of equally likely possible outcomes.

P(event)=favorable outcomestotal outcomes\begin{align*}& P \text{(event)} = \frac{\text{favorable outcomes}}{\text{total outcomes}}\end{align*}

Favorable outcomes are the outcomes you are looking for. In this case the favorable outcomes are outcomes that have a sum of 5. To find the number of total outcomes for the two tosses, you can use the Counting Principle. Since each toss of a number cube has 6 different outcomes.

total outcomes=6 outcomes6 outcomes=36 total outcomes\begin{align*} \text{total outcomes} &= 6 \ \text{outcomes} \cdot 6 \ \text{outcomes}\\ &= 36\ \text{total outcomes}\end{align*}

To figure out this particular problem we must add another step. Now list those 36 outcomes and mark the outcomes that result in a sum of 5.

\begin{align*}1-1 \quad 2-1 \quad 3-1 \quad {\color{red}\mathbf{4-1}} \quad 5-1 \quad 6-1 \\ 1-2 \quad 2-2 \quad {\color{red}\mathbf{3-2}} \quad 4-2 \quad 5-2 \quad 6-2 \\ 1-3 \quad {\color{red}\mathbf{2-3}} \quad 3-3 \quad 4-3 \quad 5-3 \quad 6-3 \\ {\color{red}\mathbf{1-4}} \quad 2-4 \quad 3-4 \quad 4-4 \quad 5-4 \quad 6-4 \\ 1-5 \quad 2-5 \quad 3-5 \quad 4-5 \quad 5-5 \quad 6-5 \\ 1-6 \quad 2-6 \quad 3-6 \quad 4-6 \quad 5-6 \quad 6-6 \end{align*}

Since there are 4 outcomes that have a sum of 5:

\begin{align*}P(5) = \frac{4}{36}=\frac{1}{9}\end{align*}

The probability of rolling two number cubes with a sum of 5 is \begin{align*}\frac{1}{9}\end{align*}.

IV. Find Probabilities of Specified Outcomes Using the Counting Principle

In the last section, we started using the Counting Principle to calculate probability. Let’s explore this a little further.

Example

What is the probability of spinning the spinner twice and having it land on the same color both times?

Step 1: Rather than draw a tree diagram, use the Counting Principle to find the number of total outcomes. Since each spin has 4 outcomes:

\begin{align*} \text{total outcomes}&= 4 \ \text{outcomes} \cdot 4 \ \text{outcomes}\\ &= 16 \ \text{total outcomes}\end{align*}

Step 2: Now list all 16 outcomes and find the number of ways both spinners can land on the same color both times:

\begin{align*}&{\color{red} \mathbf{red - red}} && \text{blue}-\text{red} && \text{yellow}-\text{red} && \text{green}-\text{red} \\ &\text{red}-\text{blue} && {\color{red} \mathbf{blue - blue}} && \text{yellow}-\text{blue} && \text{green}-\text{blue}\\ &\text{red}-\text{yellow} && \text{blue}-\text{yellow} && {\color{red} \mathbf{yellow - yellow}} && \text{green}-\text{yellow}\\ &\text{red}-\text{green} && \text{blue}-\text{green} && \text{yellow}-\text{green} && {\color{red} \mathbf{green - green}}\end{align*}

Step 3: Find the ratio of favorable outcomes to total outcomes:

\begin{align*} P \text{(same)} = \frac{favorable}{total \ outcomes}=\frac{4}{16}= \frac{1}{4}\end{align*}

The probability of the arrow landing on the same color two times in a row is \begin{align*}\frac{1}{4}\end{align*}.

Now let’s go back and apply what we have learned to the problem from the introduction.

## Real-Life Example Completed

The Bike Shop

Here is the original problem once again. Reread it and then figure out the total number of options that Telly has.

Telly and Carey are working in a bicycle shop over the spring vacation. They are excited because Ms. Kelley the owner is going to let each of them choose a new bike when the week is done. This way, they get to work and earn a new bike at the same time.

There are so many different types of bikes, that Telly is having a difficult time choosing. To complicate matters even more, Ms. Kelley said that she would let Telly design her own bike so she can choose seat color and type of handlebars as well as color of the bike. Telly knows that she wants a mountain bike, so at least that part has been chosen. Here are all of the options that Telly has.

Mountain bike

Colors = Red, Green, Blue or Purple

Seat = normal or extra cushion

Handlebars = straight or curved

Telly took out a piece of paper and tried drawing out all of her options. She became frustrated almost immediately.

You can use the Counting Principle or a tree diagram to figure out the outcomes.

Solution to Real – Life Example

First, list out the options once again.

Mountain bike

Colors = Red, Green, Blue or Purple

Seat = normal or extra cushion

Handlebars = straight or curved

Next, we can use the Counting Principle to calculate the total number of options that Telly has for her bike.

There are four colors = 4

There are two seat options = 2

There are two handlebar options = 2

\begin{align*}4 \times 2 \times 2 = 16 \end{align*} possible bike options

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Probability
the mathematical way of calculating the likelihood of an event occurring, the ratio of favorable outcomes to total outcomes.
Event
a result of an experiment or activity.
Outcome
a possible result of an event occurring.
Total Outcomes
all possible outcomes
Tree Diagram
a branching diagram that shows all possible outcomes for an event.
Counting Principle
\begin{align*} \text{Outcomes} \times \text{outcomes} = \text{total outcomes}\end{align*}
Favorable Outcomes
The outcomes that you are looking for.

## Time to Practice

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

1. 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?
2. 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?
3. 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? Use a tree diagram to list them all.

4. 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? Use a tree diagram to list them all.
5. 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? List the outcomes using a tree diagram.

Directions: Use the counting principle to determine the number of outcomes.

1. Nigel dropped 3 open-faced peanut butter sandwiches that were equally likely to land face-up or face-down. How many different outcomes are there?
2. Movie Star Toothpaste comes in 3 different flavors: sparkle, blast, and stripe, 3 different sizes, and 2 different tube styles. How many toothpaste choices are there?
3. Dave made 3 predictions for this Sunday’s football games. How many different outcomes of being right or wrong are there? List the outcomes.
4. Bridget made 4 predictions for this Sunday’s football games. How many different outcomes of being right or wrong are there? List the outcomes.
5. How many more outcomes will there be if Bridget adds an extra prediction for problem 4 above?
6. Sandy’s Sandals come in 4 different models: sport, super-sport, casual, and chic, 5 different colors, and 9 different sizes. How many choices are there?
7. In problem 6 above, Sandy’s has run out of size 8 sandals of all types. How many fewer choices are there?
8. Mike’s Bikes features three different bike styles: mountain, racer, and stunt bike. You can choose from 6 different gear systems, 4 frame alloys, and 5 colors. How many bike choices are there?
9. In problem 8 above, Mike gets a new bike color but now features only 5 different gear systems. Does he have more or fewer choices now? How many more or fewer?
10. Mavis and Marvin each have a deck of 52 cards. Each chooses 1 card from their deck. How many different outcomes are possible for the two events?
11. Tilly spins a spinner that has red, blue, and yellow sections 3 times. How many different outcomes are possible?
12. Tilly spins a spinner that has red, blue, and yellow sections and tosses a number cube. How many different outcomes are possible?
13. Tilly tosses a number cube 2 times. How many different outcomes are possible?
14. Tilly tosses a number cube 3 times. How many different outcomes are possible?

Directions: Use the counting principle to determine probabilities.

1. Kalyani spins the spinner 2 times. What is the probability that the arrow will land on green both times?
2. Kalyani spins the spinner 2 times. What is the probability that the arrow will land on the same color both times?
3. Kalyani spins the spinner 2 times. What is the probability that the arrow will land on a different color each time?
4. Kalyani spins the spinner 3 times. What is the probability that the arrow will land on a different color each time?
5. Kalyani spins the spinner 3 times. What is the probability that red will come up at least one time?
6. Kalyani spins the spinner 3 times. What is the probability that red will come up 2 or more times?

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