Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in Figure 1. They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

### Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is $\left\{1,2,3,4,5,6\right\}.$ An event is any subset of a sample space.

The likelihood of an event is known as probability. The probability of an event $p$ is a number that always satisfies $0\le p\le 1,$ where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like Table 1.

Outcome | Probability |
---|---|

Winning the raffle | 1% |

Losing the raffle | 99% |

The sum of the probabilities listed in a probability model must equal 1, or 100%.

### How To

**Given a probability event where each event is equally likely, construct a probability model.**

- Identify every outcome.
- Determine the total number of possible outcomes.
- Compare each outcome to the total number of possible outcomes.

### Example 1

#### Constructing a Probability Model

Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.

### Q&A

**Do probabilities always have to be expressed as fractions?**

*No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1. *

Construct a probability model for tossing a fair coin.

### Computing Probabilities of Equally Likely Outcomes

Let $S$ be a sample space for an experiment. When investigating probability, an event is any subset of $S.$ When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in $S.$ Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in $S,$ so the probability of the event is $\frac{4}{6}=\frac{2}{3}.$

### Computing the Probability of an Event with Equally Likely Outcomes

The probability of an event $E$ in an experiment with sample space $S$ with equally likely outcomes is given by

$E$ is a subset of $S,$ so it is always true that $0\le P(E)\le 1.$

### Example 2

#### Computing the Probability of an Event with Equally Likely Outcomes

A six-sided number cube is rolled. Find the probability of rolling an odd number.

A number cube is rolled. Find the probability of rolling a number greater than 2.

### Computing the Probability of the Union of Two Events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events $E\phantom{\rule{0.8em}{0ex}}\text{and}F,\text{written}E\cup F,$ is the event that occurs if either or both events occur.

Suppose the spinner in Figure 2 is spun. We want to find the probability of spinning orange or spinning a $b.$

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is $\frac{3}{6}=\frac{1}{2}.$ There are a total of 6 sections, and 2 of them have a $b.$ So the probability of spinning a $b$ is $\frac{2}{6}=\frac{1}{3}.$ If we added these two probabilities, we would be counting the sector that is both orange and a $b$ twice. To find the probability of spinning an orange or a $b,$ we need to subtract the probability that the sector is both orange and has a $b.$

The probability of spinning orange or a $b$ is $\frac{2}{3}.$

### Probability of the Union of Two Events

The probability of the union of two events $E$ and $F$ (written $E\cup F$ ) equals the sum of the probability of $E$ and the probability of $F$ minus the probability of $E$ and $F$ occurring together $\text{(}$ which is called the intersection of $E$ and $F$ and is written as $E\cap F$ ).

### Example 3

#### Computing the Probability of the Union of Two Events

A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

### Computing the Probability of Mutually Exclusive Events

Suppose the spinner in Figure 2 is spun again, but this time we are interested in the probability of spinning an orange or a $d.$ There are no sectors that are both orange and contain a $d,$ so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is

Notice that with mutually exclusive events, the intersection of $E$ and $F$ is the empty set. The probability of spinning an orange is $\frac{3}{6}=\frac{1}{2}$ and the probability of spinning a $d$ is $\frac{1}{6}.$ We can find the probability of spinning an orange or a $d$ simply by adding the two probabilities.

The probability of spinning an orange or a $d$ is $\frac{2}{3}.$

### Probability of the Union of Mutually Exclusive Events

The probability of the union of two *mutually exclusive* events $E\phantom{\rule{0.8em}{0ex}}\text{and}\phantom{\rule{0.8em}{0ex}}F$ is given by

### How To

**Given a set of events, compute the probability of the union of mutually exclusive events.**

- Determine the total number of outcomes for the first event.
- Find the probability of the first event.
- Determine the total number of outcomes for the second event.
- Find the probability of the second event.
- Add the probabilities.

### Example 4

#### Computing the Probability of the Union of Mutually Exclusive Events

A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.

A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

### Using the Complement Rule to Compute Probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will *not* happen. The **complement of an event** $E,$ denoted ${E}^{\prime},$ is the set of outcomes in the sample space that are not in $E.$ For example, suppose we are interested in the probability that a horse will lose a race. If event $W$ is the horse winning the race, then the complement of event $W$ is the horse losing the race.

To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.

The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is $\frac{1}{9},$ the probability of the horse losing the race is simply

### The Complement Rule

The probability that the complement of an event will occur is given by

### Example 5

#### Using the Complement Rule to Calculate Probabilities

Two six-sided number cubes are rolled.

- ⓐFind the probability that the sum of the numbers rolled is less than or equal to 3.
- ⓑFind the probability that the sum of the numbers rolled is greater than 3.

Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.

### Computing Probability Using Counting Theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.

Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are $C(5,2)$ ways to select 2 phones that are not defective. There are 8 phones, so there are $C(8,2)$ ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

### Example 6

#### Computing Probability Using Counting Theory

A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears.

- ⓐFind the probability that only bears are chosen.
- ⓑFind the probability that 2 bears and 3 dogs are chosen.
- ⓒFind the probability that at least 2 dogs are chosen.

A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs.

- ⓐFind the probability that all 3 gumballs selected are purple.
- ⓑFind the probability that no yellow gumballs are selected.
- ⓒFind the probability that at least 1 yellow gumball is selected.

### Media

Access these online resources for additional instruction and practice with probability.

### 11.7 Section Exercises

#### Verbal

What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.

What is a sample space?

What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times.

The *union of two sets* is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the *union of two events* from a probability model? How is it different?

#### Numeric

For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated.

Landing on red

Not landing on blue

Landing on blue or a vowel

Landing on yellow or a consonant

For the following exercises, two coins are tossed.

What is the sample space?

Find the probability of tossing exactly one tail.

For the following exercises, four coins are tossed.

What is the sample space?

Find the probability of tossing exactly three heads.

Find the probability of tossing all tails.

Find the probability of tossing exactly two heads or at least two tails.

For the following exercises, one card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

A club

Six or seven

An ace or a diamond

A heart or a non-jack

For the following exercises, two dice are rolled, and the results are summed.

Find the probability of rolling a sum of $3.$

Find the probability of rolling an odd sum less than $9.$

Find the probability of rolling a sum less than $15.$

Find the probability of rolling a sum between $6$ and $9\text{,}$ inclusive.

Find the probability of rolling any sum other than $5$ or $6.$

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:

A tail on the coin or red ace

No aces

For the following exercises, use this scenario: a bag of M&Ms contains $12$ blue, $6$ brown, $10$ orange, $8$ yellow, $8$ red, and $4$ green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

What is the probability of getting $4$ blue M&Ms?

What is the probability of getting no brown M&Ms?

#### Extensions

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting $20$ numbers from the numbers $1$ to $80.$ After the player makes his selections, $20$ winning numbers are randomly selected from numbers $1$ to $80.$ A win occurs if the player has correctly selected $3,4,$ or $5$ of the $20$ winning numbers. (Round all answers to the nearest hundredth of a percent.)

What is the percent chance that a player selects exactly 4 winning numbers?

What is the percent chance of winning?

How much less is a player’s chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers?

#### Real-World Applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).^{35}

If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true?