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Applications of Probability

Examples of probability and statistics being used in everyday life.

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Calculate Odds Using Outcomes or Probability
License: CC BY-NC 3.0

When you think about chances and odds, you can calculate the likelihood that an event will occur or not occur.

Jamie was listening to the weather forecast to determine if he and his friends should go to the beach. The weather forecaster said that there was a 4 in 5 chance of showers. What is the chance that he will be going to the beach with his friends? How can Jamie express his odds as a fraction and a percent?

In this concept, you will learn to calculate odds by using outcomes or probability.

Outcomes

The probability of an event is defined as a ratio that compares the favorable outcomes to the total outcomes. This ratio can be expressed in fraction form.

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

Sometimes people express the likelihood of events in terms of odds rather than probabilities. The odds of an event occurring are equal to the ratio of favorable outcomes to unfavorable outcomes.

\begin{align*}\text{(odds)} = \frac{ \# \ \text{of favorable outcomes}}{\text{unfavorable} \ \# \ \text{of outcomes}}\end{align*}

License: CC BY-NC 3.0

Think about the odds for the arrow of the spinner above landing on red:

Favorable outcomes = 1 (red) 

Unfavorable outcomes = 2 (blue, yellow)

Total outcomes = 3

So the probability of spinning red is:

\begin{align*}\begin{array}{rcl} P \ \text{(red)} &=& \frac{\# \ \text{of favorable outcomes}}{\ \text{total # of outcomes}} \\ P \ \text{(red)} &=& \frac{1}{3} \end{array}\end{align*}

While the odds in favor of red are:

\begin{align*}\begin{array}{rcl} \text{odds} \ \text{(in favor of red)} &=& \frac{ \# \ \text{of favorable outcomes}}{\# \ \text{of unfavorable outcomes}} \\ \text{odds} \ \text{(in favor of red)} &=& \frac{1}{2} \end{array}\end{align*}

Odds against an event occurring are defined as:

\begin{align*}\begin{array}{rcl} \text{odds} \ (\text{against red)} &=& \frac{ \# \ \text{of unfavorable outcomes}}{\# \ \text{of favorable outcomes}} \\ \text{odds} \ (\text{against red)} &=& \frac{2}{1} \end{array}\end{align*}

You can solve any probability problem in terms of odds rather than probabilities. Notice that the ratio represents what is being compared. Be sure that your numbers match the comparison.

You can use odds to calculate how likely an event is to happen. You can compare the odds in favor of an event with the probability that the event will actually occur.

Let’s look at an example.

Suppose the weather forecast states:

  • Odds in favor of rain: 7 to 3

These odds tell you not only the odds of rain, but also the odds of not raining.

If the odds in favor or rain are 7 to 3, then the odds against rain are:

  • Odds against rain: 3 to 7

Another way of saying that is:

  • Odds that it will NOT rain: 3 to 7

You can use this idea in many different situations. If you know the odds that something will happen, then you also know the odds that it will not happen.

Examples

Example 1

Earlier, you were given a problem about Jamie’s rainy beach day.

Jamie is trying to determine if he should go to the beach with his friends and is listening to the weather forecast.

First, what is the ratio from the question?

Forecast: 4 in 5 chance of showers.

Therefore it will rain 4 times out of 5.

The chances of it not raining are 1 in 5.

Next, write the ratio of favorable to total outcomes.

\begin{align*}\begin{array}{rcl} P \text{(raining)} &=& \frac{ \# \ \text{of favorable outcomes}}{\text{total # of outcomes}} \\ P \text{(raining)} &=& \frac{4}{5} \end{array}\end{align*}

Then express this fraction as a percent.

\begin{align*}\begin{array}{rcl} \frac{4}{5} &=& \frac{x}{100} \\ 5x &=& 4 \times 100 \\ 5x &=& 400 \\ \frac{5x}{5} &=& \frac{400}{5} \\ x &=& 80 \end{array}\end{align*}

The answer is 80.

There is an 80% chance of rain.

Example 2

What are the odds in favor of a number cube landing on 4?

First, find the favorable and unfavorable outcomes.

Favorable outcomes = 1(4)  

Unfavorable outcomes = 5(1, 2, 3, 5, 6)

Next, write the ratio of favorable to unfavorable outcomes.

\begin{align*}\begin{array}{rcl} \text{odds} \ (4) &=& \frac{ \# \ \text{of favorable outcomes}}{\# \ \text{of unfavorable outcomes}} \\ \text{odds} \ (4) &=& \frac{1}{5} \end{array}\end{align*}

The answer is \begin{align*}\frac{1}{5}\end{align*}.

The odds in favor of rolling a 4 are 1 to 5.

Use this spinner to calculate odds.

License: CC BY-NC 3.0

Example 3

Calculate the odds in favor of spinning a blue.

First, find the favorable and unfavorable outcomes.

Favorable outcomes = 1 (blue) 

Unfavorable outcomes = 2 (red, yellow)

Next, write the ratio of favorable to unfavorable outcomes.

\begin{align*}\begin{array}{rcl} \text{odds} \ \text{(blue)} &=& \frac{ \# \ \text{of favorable outcomes}}{\# \ \text{of unfavorable outcomes}} \\ \text{odds} \ \text{(blue)} &=& \frac{1}{2} \end{array}\end{align*}

The answer is \begin{align*} \frac{1}{2}\end{align*}.

The odds in spinning a blue are 1 to 2.

Example 4

Calculate the odds in favor of spinning a red or blue.

First, find the favorable and unfavorable outcomes.

Favorable outcomes = 2 (blue or red) 

Unfavorable outcomes = 1 (yellow)

Next, write the ratio of favorable to unfavorable outcomes.

\begin{align*}\begin{array}{rcl} \text{odds} \ \text{(blue or red)} &=& \frac{ \# \ \text{of favorable outcomes}}{\# \ \text{of unfavorable outcomes}} \\ \text{odds} \ \text{(blue or red)} &=& \frac{2}{1} \end{array}\end{align*}

The answer is \begin{align*}\frac{2}{1}\end{align*}.

The odds in spinning a blue or a red are 2 to 1.

Example 5

Calculate the odds against spinning a red or blue.

First, find the favorable and unfavorable outcomes.

Favorable outcomes = 2 (blue or red)

Unfavorable outcomes = 1 (yellow)

Next, write the ratio of favorable to unfavorable outcomes.

\begin{align*}\begin{array}{rcl} \text{odds} \ \text{(against red or blue)} &=& \frac{ \# \ \text{of unfavorable outcomes}}{\# \ \text{of favorable outcomes}} \\ \text{odds} \ \text{(against red or blue)} &=& \frac{1}{2} \end{array}\end{align*}

The answer is \begin{align*}\frac{1}{2}\end{align*}.

The odds against spinning a blue or a red are 1 to 2.

Review

1. For rolling a number cube, what are the odds in favor of rolling a 2?

2. For rolling a number cube, what are the odds against rolling a 2?

3. For rolling a number cube, what are the odds in favor of rolling a number greater than 3?

4. For rolling a number cube, what are the odds in favor rolling a number less than 5?

5. For rolling a number cube, what are the odds against rolling a number less than 5?

6. For rolling a number cube, what are the odds in favor of rolling an even number?

7. For rolling a number cube, what are the odds against rolling an even number?

For a spinner numbered 1 –10, answer the following questions.

8. For spinning the spinner, what are the odds in favor of the arrow landing on 10?

9. For spinning the spinner, what are the odds in favor of the arrow landing on a 2 or 3?

10. For spinning the spinner, what are the odds in favor of the arrow landing on 7, 8 or 9?

11. For spinning the spinner, what are the odds in favor of NOT landing on an even number?

12. For spinning the spinner, what are the odds of the arrow NOT landing on 10?

13. For spinning the spinner, what are the odds in favor of the arrow landing on a number greater than 2?

14. For spinning the spinner, what are the odds in favor of the arrow NOT landing on a number greater than 2?

15. For spinning the spinner, what are the odds of the arrow not landing on a number greater than 3?

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.11. 

Resources

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Vocabulary

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.

Complementary Events

Complementary events can occur in a single trial of a given experiment.

consumer

A consumer is anyone who purchases or uses a particular product or service.

counter-intuitive

If something is counterintuitive, it is not what you might initially guess.

Data

Data is information that has been collected to represent real life situations, usually in number form.

demographic

A demographic is a specific group of consumers chosen by age, sex, religion, home country, place of residence, music preference, or any other specified characteristic.

Disjoint Events

Disjoint or mutually exclusive events cannot both occur in a single trial of a given experiment.

fair die

A fair die is a die with an equal chance of landing on any side.

Fundamental Counting Principle

The Fundamental Counting Principle states that if an event can be chosen in p different ways and another independent event can be chosen in q different ways, the number of different arrangements of the events is p x q.

Permutation

A permutation is an arrangement of objects where order is important.

Random

When everyone or everything in a population has an equal chance of being selected, the selection can be said to occur at random.

sample point

A sample point is just one of the possible outcomes in a sample.

Target Audience

A target audience is a particular demographic that is intended to be the main group of consumers of a particular product or service.

unfair die

An unfair die would be more likely to land on a particular number than the others.

variable

In statistics, a variable is simply a characteristic that is being studied.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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