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# Definition of Probability

## Probability = # of Favorable Outcomes/# of Possible Outcomes

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Definition of Probability
Credit: Jeannot Doe
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Mrs. Woodfaulk uses colored balls to demonstrate the concept of probability. She places 1 red ball, 1 blue, 3 yellow balls, and 4 green balls in a bag and closes the bag. She places three stacks of colored tokens on her desk. One stack is red. One is yellow. And one stack is green. Then, Mrs. Woodfaulk tells the class that she is going to pull one ball out the bag, and she wants each student to guess what color the ball will be by choosing the same color token. How can Rachel write a statement of probability for each color ball?

In this concept, you will learn the definition of probability and how to write statements of probability.

### Writing Statements of Probability

Probability is the measure of the likeliness that an event will occur. Probability is used 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, you are still talking about probability.

A ratio is used to calculate probability. A ratio is a way of comparing two quantities. With probability, you can compare the number of favorable outcomes to the amount of possible outcomes.

Here is a ratio.

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

Notice that the ratio is in fraction form. This is one way to figure out the probability of an event happening.

Here is an example to illustrate the concept of probability. As you read this example, think about the number of possible outcomes first. That is your denominator. Then calculate the number of favorable outcomes, your numerator.

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

First, determine the number of possible outcomes. Since the number cube is numbered 1 – 6, there are only 6 possible outcomes. That is the denominator.

\begin{align*}P = \frac{\text{# number of favourable outcomes}}{6}\end{align*}

Next, consider the number of favorable outcomes. Since 2 is the only favorable outcome, there is only 1 favorable outcome. That is the numerator.

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

This is the probability of Mark rolling a 2.

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?

First, the number of possible outcomes did not change. It is still a 6.

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

Next, determine the number of favorable outcomes. The questioned asked about spinning an odd number. Counting from 1 – 6, there are three odd numbers. Therefore, the number of favorable outcomes is 3.

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

Then, simplify the probability if necessary.

### Examples

#### Example 1

Earlier, you were given a problem about Rachel and Mrs. Woodfaulk’s probability demonstration.

Mrs. Woodfaulk placed 1 red ball, 1 blue, 3 yellow balls, and 4 green balls in the bag, and then she asked the students to choose a token that is the color of the ball she will randomly pull from the bag. The students in the first three rows chose their tokens, and most of them chose green tokens. A few chose yellow tokens. And none of them chose the red token. Rachel was planning to choose a yellow token, but as she waited in line, she began to rethink her decision.

Should Rachel reconsider her decision to choose a yellow token? How can she write a probability statement that will help her choose the most likely colored ball?

First, determine the number of possible outcomes. Since there are a total of 9 colored balls in the bag, there are 9 possible outcomes. That is the denominator.

\begin{align*}P = \frac{}{9}\end{align*}

Next, determine which color ball will have the highest number of favorable outcomes. There are 4 green balls in the bag, so the number of favorable outcomes for green is 4. That is the numerator.

\begin{align*}P = \frac{4}{9}\end{align*}

The probability of Mrs. Woodfaulk pulling a green ball is four out of nine. The probability of pulling a yellow ball is 3 out of nine. And the probability of pulling a red ball or a blue ball out of the bag is 1 out of 9.

So, Rachel should choose a green token because the probability is more likely that Mrs. Woodfaulk will pull a green ball from the bag.

#### Example 2

Find the probability.

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?

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.

\begin{align*}P = \frac{}{8}\end{align*}

Next, determine the number of favorable outcomes or all of the possibilities that are not yellow or red. This means you count the green and the blue squares. There is one green and one blue square, so the number of favorable outcomes is 2. That is the numerator.

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

Then, simplify the fraction by dividing the numerator and denominator by their greatest common factor, 2.

\begin{align*}\begin{array}{rcl} 2 \div 2 &=& 1 \\ 8 \div 2 &=& 4 \end{array}\end{align*}

This becomes

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

Write a ratio to show the probability for each question below regarding this scenario.

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

#### Example 3

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

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.

\begin{align*}P = \frac{}{8}\end{align*}

Next, determine the number of favorable outcomes. This means count the red squares. There are 2, so the number of favorable outcomes is 2. That is the numerator.

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

Then, simplify the fraction by dividing the numerator and denominator by their greatest common factor, 2.

\begin{align*}\begin{array}{rcl} 2 \div 2 &=& 1 \\ 8 \div 2 &=& 4 \end{array}\end{align*}

This becomes

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

#### Example 4

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

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.

\begin{align*}P = \frac{}{8}\end{align*}

Next, determine the number of favorable outcomes or yellow squares. There are four yellow squares, so the number of favorable outcomes is 4. That is the numerator.

\begin{align*}P = \frac{4}{8}\end{align*}

Then, simplify the fraction by dividing the numerator and denominator by their greatest common factor, 4.

\begin{align*}\begin{array}{rcl} 4 \div 4 &=& 1 \\ 8 \div 2 &=& 4 \end{array}\end{align*}

This becomes

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

#### Example 5

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

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.

\begin{align*}P = \frac{}{8}\end{align*}

Next, determine the number of favorable outcomes or the number of yellow and blue cubes. There are 4 yellow squares and 1 blue square, so the number of favorable outcomes is 5. That is the numerator.

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

This is the answer and it cannot be simplified.

### Review

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 based on the scenario above.

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

To see the Review answers, open this PDF file and look for section 12.14.

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