# 12.14: Definition of Probability

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

**Practice**Definition of Probability

At the amusement park, Taylor watched a group of people play a shell game. In the game, the operator had three shells and he put a ball under one of the shells. Then the shells were switched around and each player tried to guess where the ball was located.

Taylor watched for a while, but still couldn't seem to figure out the game. She knew it had to do with probability.

What is the probability of the blue ball being under one of the shells?

Do you know?

Do you know how to write this as a statement of probability?

**This Concept is about defining and understanding probability. By the end of it, you will know how to answer this question.**

### Guidance

**We use probability all the time in real life situations. If you watch the weather in the morning you may hear the meteorologist talk about a 20% chance or rain or snow. In this case a percentage gives us the probability that it would rain. While there is a 20% chance that it will rain, there is an 80% chance that it won’t rain. All in all, we are still talking about probability.**

**How can we calculate probability?**

To calculate probability we use a ** ratio**. If you remember back to earlier Concepts, you will remember that a ratio is a way of comparing two quantities. With probability, we can compare the number of favorable outcomes to the amount of possible outcomes.

Here is our ratio.

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

**Notice that the ratio is in fraction form. That is one way that we can compare to figure out the probability of an event happening.**

**How can we apply this ratio?**

To apply this ratio, we have to look at an example. As you read this example, think about the number of possible outcomes first. That is our denominator. Then go to the number of favorable outcomes.

Mark is rolling a number cube that is numbered 1 – 6. What are the chances that Mark will roll a 2?

**To work through this problem and figure out the probability we first need to determine the number of possible outcomes. Since the number cube is numbered 1 – 6, there are only 6 possible outcomes. That is our denominator.**

\begin{align*}P = \frac{number \ of \ favorable \ outcomes}{6}\end{align*}

**Next we think of the number of favorable outcomes. Since we are only looking for a two, there is one favorable outcome. That is our numerator.**

\begin{align*}P = \frac{1}{6}\end{align*}

That one was an introductory problem. Now let’s look at one that is a little more complicated.

Jessie spins the same number cube. She wants to spin an odd number. What are the chances that she will spin an odd number?

**Let’s break this one down. First, the number of possible outcomes did not change. It is still a 6.**

\begin{align*}P = \frac{\# \ of \ Favorable \ outcomes}{6}\end{align*}

**The number of favorable outcomes did change. We are looking for an odd number. If we count from 1 – 6, there are three odd numbers. Therefore, the number of favorable outcomes is 3.**

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

**Notice that we can simplify the probability too. Sometimes that will give an even clearer picture of the likelihood that the event will or will not happen.**

Practice finding probability. Write a ratio to show the probability for each question below.

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

#### Example A

What is the probability of Jake pulling out a red cube?

**Solution:\begin{align*} \frac{2}{8}\end{align*}**

#### Example B

What is the probability of Jake pulling out a yellow cube?

**Solution: \begin{align*}\frac{4}{8}\end{align*}**

#### Example C

What is the probability of Jake pulling out a yellow or blue cube?

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

Now back to Taylor and the shell game. Here is the original problem once again.

At the amusement park, Taylor watched a group of people play a shell game. In the game, the operator had three shells and he put a ball under one of the shells. Then the shells were switched around and each player tried to guess where the ball was located.

Taylor watched for a while, but still couldn't seem to figure out the game. She knew it had to do with probability.

What is the probability of the blue ball being under one of the shells?

Do you know?

Do you know how to write this as a statement of probability?

To write this as a probability, we look at the chances.

There is a one ball.

There are three shells.

There is a one in three chance of finding the ball.

This is the probability. We can write it as a fraction.

**Our answer is \begin{align*}\frac{1}{3}\end{align*}.**

### 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 yellow or a red square?

**Answer**

To figure out this probability, we include all of the possibilities that are not yellow or red. This means we count the green and the blue squares.

\begin{align*}\frac{2}{8}\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

Khan Academy, Basic 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. Write a fraction to show the following probabilities.

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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Here you'll learn to define and understand probability.