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# 11.5: Probability and Odds

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Rainstorm

Telly and Carey were already hard at work when Ms. Kelley came into the bike shop on Thursday morning. It was three days before the big race and there was still a lot of work to be done.

“I can’t believe it!” Ms. Kelley exclaimed as she came into the shop.

“There is a 4 to 5 chance that it is going to rain on Saturday. I just heard the weather report,” Ms. Kelley said sighing.

“Well, there is still a chance that it won’t,” Telly said trying to cheer her up.

When we think about chances and odds, we can calculate the likelihood that an event will or won’t occur. In this case, there are odds that it will rain and odds that it won’t. We can also express those odds as a fraction or a percentage. Learn about odds in this lesson and you can work on the odds of the rainstorm at the end.

What You Will Learn

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

• Recognize and distinguish among overlapping, disjoint and complementary events, including the use of Venn diagrams and set notation.
• Recognize the odds in favor of events as the ratio of favorable outcomes to unfavorable outcomes.
• Recognize the odds in favor of events are the ratio of the probability that the event will occur.
• Calculate odds using outcomes or probabilities.
• Calculate probabilities given odds.

Teaching Time

I. Recognize and Distinguish among Overlapping, Disjoint and Complementary Events

When we spin a spinner or roll a dice to calculate a probability, some probabilities have events in common and some don’t. This is where we can begin to talk about identifying disjoint events. Disjoint events are events that don’t have any outcomes in common.

Example

Consider spinning this spinner.

• Event Y: {yellow}
• Event B: {blue}

Events A and B are disjoint events because they have no outcomes in common – the arrow either lands on blue or yellow.

We can use a Venn diagram to show when events overlap and when they don’t overlap. A Venn diagram is something that you may have seen before. It has round shapes that overlap or don’t overlap. The Venn diagram for disjoint events shows no overlap between the two events.

Not all events are disjoint. There are many events that are connected to each other. Let’s look at the following example.

Example

Consider this spinner and the events R (red) and T (top).

• Event R: {red-top, red-bottom}
• Event T: {red-top, blue-top}

Clearly, both events share an outcome – red-top – so the two are called overlapping events. The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes.

Complementary events are events whose probability sum adds up to 1 (decimal) or 100 percent.

Events G and B in this spinner above are complementary.

\begin{align*}P \text{(yellow)} + P \text{(green)} = 1\end{align*}

Complementary events are either-or events. Either the spinner above lands on green or it lands on yellow. There are no other outcomes.

II. Recognize the Odds in Favor of Events as the Ratio of Favorable Outcomes to Unfavorable Outcomes

You’ve seen that the probability of an event is defined as a ratio that compares the favorable out comes to the total outcomes. We can write this ratio in fraction form.

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

For example, for the arrow of the spinner above landing on red:

\begin{align*}\text{favorable outcomes} &= 1 \text{(red)} \\ \text{unfavorable outcomes} &= 2 \text{(blue, yellow)} \\ \text{total outcomes} &= 3\end{align*}

So the probability of spinning red is:

\begin{align*}P \text{(red)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{3}\end{align*}

While the odds in favor of red are:

\begin{align*}\text{Odds(in favor of red)} = \frac{favorable \ outcomes}{unfavorable \ outcomes} = \frac {1}{2}\end{align*}

Odds against an event occurring are defined as:

\begin{align*} \text{Odds(against red)} = \frac{unfavorable \ outcomes}{favorable \ outcomes} = \frac{2}{1} \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.

Example

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

Step 1: Find the favorable and unfavorable outcomes.

\begin{align*}\text{favorable outcomes} &= 1(4) \\ \text{unfavorable outcomes} &= 5 (1, 2, 3, 5, 6)\end{align*}

Step 2: Write the ratio of favorable to unfavorable outcomes.

\begin{align*}\text{Odds}(4) = \frac{favorable \ outcomes}{unfavorable \ outcomes} = \frac{1}{5}\end{align*}

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

III. Recognize the Odds in Favor of Event as the Ratio of the Probability that the Event Will Occur

We can use odds to calculate how likely an event is to happen. We can compare the odds in favor of an event with the probability that the event will actually occur. Let’s look at an example.

Example

You’ve seen that the odds in favor of an event \begin{align*}(E)\end{align*} occurring are shown in this ratio.

\begin{align*}\text{Odds(in favor of} \ E) = \frac{favorable \ outcome}{unfavorable \ outcome} = \frac{1}{2}\end{align*}

And the odds against the same event occurring are:

\begin{align*}\text{Odds(against} \ E) = \frac{unfavorable \ outcome}{favorable \ outcome} = \frac{2}{1}\end{align*}

You can use these two facts to compute the ratio of things happening and not happening.

For 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.

IV. Calculate Odds using Outcomes or Probabilities

You can express the likelihood of an event using odds or probabilities. To convert probabilities into odds, go over the problem below.

Example

The weather report forecasts that there is a 30 percent probability that it will rain tomorrow. What are the odds that it will rain tomorrow?

Step 1: Turn the probability into a fraction in simplest form.

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

Step 2: Express the fraction in terms of favorable outcomes and total outcomes.

\begin{align*}P \text{(rain)} = \frac{favorable \ outcomes} {total \ outcomes} = \frac{3}{10} \end{align*}

Step 3: Subtract the favorable outcomes from the total number of outcomes to find the unfavorable outcomes.

\begin{align*} \text{total outcomes} - \text{favorable outcomes} &= \text{unfavorable outcomes} \\ 10 - 3 &= 7 \end{align*}

Step 4: Use the favorable outcomes and unfavorable outcomes to find the odds.

\begin{align*}\text{Odds(rain)} = \frac{favorable \ outcomes}{unfavorable \ outcomes} = \frac{3}{7} \end{align*}

The odds that it will rain tomorrow are 3 to 7.

## Real-Life Example Completed

The Rainstorm

Here is the problem from the introduction. Answer each question at the end of the problem.

Telly and Carey were already hard at work when Ms. Kelley came into the bike shop on Thursday morning. It was three days before the big race and there was still a lot of work to be done.

“I can’t believe it!” Ms. Kelley exclaimed as she came into the shop.

“There is a 4 to 5 chance that it is going to rain on Saturday. I just heard the weather report,” Ms. Kelley said sighing.

“Well, there is still a chance that it won’t,” Telly said trying to cheer her up.

What are the chances that it won’t rain?

What are the odds that it will as a percentage?

What are the odds that it won’t as a percentage?

Solution to Real – Life Example

What are the chances that it won’t rain?

We know that the odds of it raining is 4 to 5. Therefore it is a 1 out of 5 chance that it won’t rain. Not very good odds.

What are the odds that it will as a percentage?

4 to 5 can be written as a percentage – 80% chance of rain.

What are the odds that it won’t as a percentage?

1 to 5 can be written as a percentage – 20% chance that it won’t rain.

## Vocabulary

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

Disjoint events
events that don’t have any outcomes in common.
Complementary events
probability that has a sum of 100%. Either/Or events are complementary events.

## Time to Practice

Directions: Solve the problems. For overlapping events, tell which events overlap.

1. For a flip of a coin, are events \begin{align*}H(\text{heads})\end{align*} and \begin{align*}T(\text{tails})\end{align*} disjoint or overlapping? Draw a Venn diagram to represent the events.
2. For a flip of a coin, are events \begin{align*}H(\text{heads})\end{align*} and \begin{align*}T(\text{tails})\end{align*} complementary or non-complementary?
3. For a single toss of a number cube, are \begin{align*}E(\text{even})\end{align*} and \begin{align*}T(3)\end{align*} disjoint events or overlapping events?
4. For a single toss of a number cube, are \begin{align*}E(\text{even})\end{align*} and \begin{align*}S(6)\end{align*} disjoint events or overlapping events?
5. For a single toss of a number cube, are \begin{align*}G3(\text{greater than} \ 3)\end{align*} and \begin{align*}O(\text{odd})\end{align*} disjoint events or overlapping events?
6. For a single toss of a number cube, are \begin{align*}E(\text{even})\end{align*} and \begin{align*}O(\text{odd})\end{align*} complementary events or non-complementary events?

1. For a single spin, are \begin{align*}B(\text{blue})\end{align*} and \begin{align*}G(\text{green})\end{align*} disjoint events or overlapping events? Draw a Venn diagram to represent the events.
2. For a single spin, are \begin{align*}G(\text{green})\end{align*} and \begin{align*}L(\text{left})\end{align*} disjoint events or overlapping events? Draw a Venn diagram to represent the events
3. For a single spin, are \begin{align*}Y(\text{yellow})\end{align*} and \begin{align*}R(\text{red})\end{align*} complementary or non-complementary events?
4. For a single spin, are \begin{align*}R(\text{right})\end{align*} and \begin{align*}L(\text{left})\end{align*} complementary or non-complementary events?
5. For a light switch, are ON and OFF disjoint or overlapping events?
6. For a light switch, are ON and OFF complementary or non-complementary events?
7. For an oven, are ON and OFF disjoint or overlapping events?

Directions: Solve the problems.

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?

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

1. For spinning the spinner, what are the odds in favor of the arrow landing on 10?
2. For spinning the spinner, what are the odds of the arrow NOT landing on 10?
3. For spinning the spinner, what are the odds in favor of the arrow landing on a number greater than 2?
4. For spinning the spinner, what are the odds in favor of the arrow NOT landing on a number greater than 2?
5. For spinning the spinner, what are the odds of the arrow not landing on a blue number greater than 5?

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