## Calculating the Odds of an Event

Here you’ll calculate odds by using outcomes or probability. Have you ever thought about the likelihood of an event happening? Take a look at this dilemma:

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 reading, and you can work on the odds of the rainstorm at the end.

## Guidance

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.

$P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}\\$

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.

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:

$P(\text{red})=\frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{1}{3}\\$

While the odds in favor of red are:

$\text{Odds(in favor of red)}=\frac{\text{favorable outcomes}}{\text{unfavorable outcomes}}=\frac{1}{2}\\$

Odds against an event occurring are defined as:

$\text{Odds(against red)}=\frac{\text{unfavorable outcomes}}{\text{favorable outcomes}}=\frac{2}{1}\\$

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.

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.

Take a look at this situation.

You’ve seen that the odds in favor of an event (E) occurring are shown in this ratio.

$\text{Odds(in favor of)}E=\frac{\text{favorable outcomes}}{\text{unfavorable outcomes}}=\frac{1}{2}\\$

And the odds against the same event occurring are:

$\text{Odds(against)}E=\frac{\text{unfavorable outcomes}}{\text{favorable outcomes}}=\frac{2}{1}\\$

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.

Use this spinner to calculate odds.

#### Example A

Odds in favor of spinning a blue.

Solution: $\frac{1}{2}\\$

#### Example B

Odds in favor of spinning a red or blue.

Solution: $\frac{2}{1}\\$

#### Example C

Odds against spinning a red or blue.

Solution: $\frac{1}{2}\\$

## Intro Problem Revisited

Now let’s go back to the dilemma from the beginning of the reading.

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.

### Guided Practice

Here is one for you to try on your own.

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

#### Step 1

Find the favorable and unfavorable outcomes.

• favorable outcomes = 1(4)
• unfavorable outcomes = 5(1,2,3,5,6)

#### Step 2

Write the ratio of favorable to unfavorable outcomes.

$\text{Odds}(4) = \frac{\text{favorable outcomes}}{\text{unfavorable outcomes}}=\frac{1}{5}\\$

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

### Vocabulary

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.

## Practice Questions

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?

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 in favor of the arrow landing on a 2 or 3?
3. For spinning the spinner, what are the odds in favor of the arrow landing on 7, 8 or 9?
4. For spinning the spinner, what are the odds in favor of NOT landing on an even number?
5. For spinning the spinner, what are the odds of the arrow NOT landing on 10?
6. For spinning the spinner, what are the odds in favor of the arrow landing on a number greater than 2?
7. For spinning the spinner, what are the odds in favor of the arrow NOT landing on a number greater than 2?
8. For spinning the spinner, what are the odds of the arrow not landing on a number greater than 3?