# 12.15: Measurement of Probability

**At Grade**Created by: CK-12

**Practice**Measurement of Probability

At the amusement park, Keith and Trevor went over to the carnival booths to try their luck at a few games. Keith played one game of “Whack a Mole” and won a ticket for an ice cream cone. Trevor threw a golf ball into a fish bowl and won a gold fish.

Then they both moved on to games of chance. After checking out several different games, they decided to play a game with a spinner. In this game you spin the spinner and whichever number you get determines the number of chances that you have. The object of the game is to use a bow and arrow to hit a target. In the autumn, the sixth grade had learned some archery and Keith had been particularly good at it.

“You’ve got this,” Trevor said supporting Keith. “You were the best one in the class.”

“Yes, but I want to spin the highest number on the spinner that I can.”

Keith and Trevor looked at the spinner. There were 10 sections on the spinner. That means that Kevin could spin anywhere from a one to a 10. If he only spun a one, then he would only get one shot at the target. If he spun a ten, then he would get 10 chances.

** Spinner courtesy of** http://etc.usf.edu/clipart/37700/37714/spinner-10_37714.htm

“What do you think my chances are of spinning an 8, 9 or 10?” Kevin asked Trevor.

“I don’t know, let me think about that. I also wonder what that chance would be as a percentage.” Trevor said.

“More importantly, what are the chances that I won’t spin an 8, 9, or 10?” Kevin mused.

**While Kevin and Trevor do their own figuring, it is time for you to learn about probability. At the end of this Concept, we will return to this problem and you can help Trevor and Kevin figure out the probability.**

### Guidance

In the last Concept, we looked at probability as a ratio in fraction form.

\begin{align*}P = \frac{\# \ of \ Favorable \ Outcomes}{\# \ of \ Possible \ Outcomes}\end{align*}

We wrote our ratios as fractions and simplified them when we could.

\begin{align*}\frac{3}{6}=\frac{1}{2}\end{align*}

**Let’s think about fractions for a minute. Fractions mean a part of a whole. Decimals and percents also mean a part of a whole. Therefore, we can write our probabilities as fractions, but we can also write them as decimals or as percents.**

Let’s practice writing the following probabilities three different ways.

A bag has four cubes in it, a red cube, two yellow cubes and one blue cube. What is the probability of drawing a red or yellow cube out of the bag?

**To work on this problem, let’s first write a fraction to show the ratio of possible outcomes and favorable outcomes. There are four cubes in the bag, so there are four possible outcomes. This is our denominator.**

\begin{align*}P & = \frac{\# \ of \ favorable \ outcomes}{\# \ of \ possible \ outcomes}\\ P &= \frac{\Box}{4}\end{align*}

Next, we need to figure out the favorable outcomes. We want a red or a yellow. There are two yellow cubes and one red cube. That means that there are three favorable outcomes.

\begin{align*}P = \frac{3}{4}\end{align*}

**Our next step is to write this as a decimal. To write \begin{align*}\frac{3}{4}\end{align*} as a decimal, we need to convert the fraction to one with a denominator that is a multiple of ten. We can create a proportion, or equal fraction with a denominator out of 100 to do this.**

\begin{align*}P = \frac{3}{4} & = \frac{\Box}{100}\\ 4 \times 25 &= 100\\ 3 \times 25 &= 75\\ P &= \frac{75}{100} \ or \ .75\end{align*}

**Now we can take the decimal and make it a percentage. If you look at the fraction out of 100 it is already clear what the percentage is. The percentage is 75% because percent means out of 100.**

**If you were working with the decimal only, then you move the decimal point two places to the right and then add the % sign. You move it two places because that is hundredths and % means out of 100.**

Practice writing the following probabilities as a decimal and a percentage.

#### Example A

\begin{align*}\frac{1}{4}\end{align*}

**Solution: \begin{align*}.25, 25%\end{align*}**

#### Example B

\begin{align*}\frac{1}{2}\end{align*}

**Solution: \begin{align*}.5, 50%\end{align*}**

#### Example C

\begin{align*}\frac{5}{8}\end{align*}

**Solution:\begin{align*}.625, 62.5%\end{align*}**

Now back to Keith and Trevor.

Here is the original problem once again. Then use what you have learned about probability to help Keith figure out his chances.

At the amusement park, Keith and Trevor went over to the carnival booths to try their luck at a few games. Keith played one game of “Whack a Mole” and won a ticket for an ice cream cone. Trevor threw a golf ball into a fish bowl and won a gold fish.

Then they both moved on to games of chance. After checking out several different games, they decided to play a game with a spinner. In this game you spin the spinner and whichever number you get determines the number of chances that you have. The object of the game is to use a bow and arrow to hit a target. In the autumn, the sixth grade had learned some archery and Keith had been particularly good at it.

“You’ve got this,” Trevor said supporting Keith. “You were the best one in the class.”

“Yes, but I want to spin the highest number on the spinner that I can.”

Keith and Trevor looked at the spinner. There were 10 sections on the spinner. That means that Keith could spin anywhere from a one to a 10. If he only spun a one, then he would only get one shot at the target. If he spun a ten, then he would get 10 chances.

** Spinner courtesy of** http://etc.usf.edu/clipart/37700/37714/spinner-10_37714.htm

“What do you think my chances are of spinning an 8, 9 or 10?” Keith asked Trevor.

“I don’t know, let me think about that. I also wonder what that chance would be as a percentage.”Trevor said.

“More importantly, what are the chances that I won’t spin an 8, 9, or 10?”Keith mused.

**First, let’s figure out what the probability is that Keith spin an 8, 9, or 10. There are 10 sections on the spinner. 10 is the denominator because it is the total number of possible outcomes.**

**8, 9, or 10 is the numerator. There are three favorable outcomes.**

\begin{align*}P = \frac{3}{10}\end{align*}

**Next, Trevor wondered what that chance would be if written as a percentage. To figure this out, we need to convert the fraction to a percent. We do this by creating an equal fraction out of 100.**

\begin{align*}\frac{3}{10}= \frac{30}{100}\end{align*}

**Keith has a 30% chance of spinning an 8, 9 or 10.**

**What about the chances of not spinning an 8, 9 or 10?**

**Well, if there is a 30% chance that he will spin one of those numbers, then there is 70% chance that he won’t.**

**You could say that the odds are against Keith spinning one of those numbers. It is unlikely that he will do so given his chances. However you never know.**

**Keith spins the spinner and spins a 6. He didn’t get the 8, 9 or 10, but he is happy with 6 chances. He warms up and aims the arrow. On the fifth try, Keith hits the bullseye of the target. He chooses a stuffed pink giraffe to take home to his sister.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Probability
- the chances that something will happen. It can be written as a fraction, decimal or percent.

- Ratio
- compares two quantities. In probability the ratio compares the number of favorable outcomes to the number of possible outcomes

### Guided Practice

Here is one for you to try on your own.

*Jake put eight colored squares into a bag. There are two reds, four yellows, one green and one blue.*

What is the probability that Jake will not pull out a red square? Write this answer as a decimal and a percentage.

**Answer**

First, we can write it as a ratio.

There are six other options besides red. Here is our ratio.

\begin{align*}\frac{6}{8}\end{align*}

Now we convert that to a decimal.

\begin{align*}6 \div 8 = .75\end{align*}

\begin{align*}.75\end{align*} becomes 75%

### Video Review

Here are videos for review.

Khan Academy, Basic Probability

James Sousa, Introduction to Probability

### Practice

Directions: A bag has the following 10 colored stones in it. There are 2 red ones, 2 blue ones, 3 green ones, 1 orange one, and 2 purple ones. In the last Concept, you wrote a fraction for each example. Now write a decimal and a percent for each probability.

1. One orange stone

2. A red stone

3. A green stone

4. A yellow stone

5. A blue stone or an orange one

6. A red one or a blue one

7. A green one or an orange one

8. A blue one or a green one

9. A blue one or a purple one

10. A purple one or a red one

11. Not purple

12. Not red

13. Not orange or purple

14. Not red or purple

15. Not orange

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