OBJECTIVES By the end of this section, I will be able to …

The probability of an outcome represents the chance or likelihood that the outcome will occur.

Building Blocks of Probability

An experiment is any activity for which the outcome is uncertain.

An outcome is the result of a single performance of an experiment.

The collection of all possible outcomes is called the sample space. We denote the sample space .

An event is a collection of outcomes from the sample space. To find the probability of an event, add up the probabilities of all the outcomes in the event.

Rules of Probability

A Different Perspective

As you read this chapter, notice that the perspective differs from that in previous chapters. Earlier, we were looking at a data set and trying to describe it graphically and numerically. Now, instead of trying to describe a data set, we are faced with an experimental situation, and our task is to calculate probabilities associated with various outcomes in the experiment.

EXAMPLE 1 Probability models

For the following three scenarios, determine whether or not a valid probability model exists:

Solution

To be a valid probability model, each scenario must meet both rules of probability.

EXAMPLE 2 Determining unusual or likely outcomes

For the following probabilities, determine the meaning of the indicated probability:

Solution

For the following probabilities, determine the meaning of the indicated probability:

(The solutions are shown in Appendix A.)

Classical Method of Assigning Probabilities

Let and denote the number of outcomes in event and the sample space , respectively. If the experiment has equally likely outcomes, then the probability of event is

EXAMPLE 3 Classical method: Single draw from a deck of cards

Find the probability of drawing an ace when drawing a single card at random from a deck of cards.

Solution

The sample space for the experiment in which a subject chooses a single card at random from a deck of cards is given in Figure 1. If the card is chosen truly at random, then it is reasonable to assume that each card has the same chance of being drawn. Each card is equally likely to be drawn, so we can use the classical method to assign probabilities.

A total of 52 outcomes are included in this sample space, so . Let be the event that an ace is drawn. Event consists of the four aces {A ♥, A♦, A♣, A♠}, so . Therefore, the probability of drawing an ace is

Find the probability of the following events when drawing a single card at random from a deck of cards:

(The solutions are shown in Appendix A.)

EXAMPLE 4 Classical method: Single fair die

Recall the town fair example (page 240). In the game, you win if you roll a 6 on a single roll of a single fair die. Find the probability of winning the game.

Solution

The sample space for a single die toss consists of six outcomes, . When the six outcomes are equally likely, we say that the die is fair. If the outcomes are not equally likely, then the die is loaded or defective. If we assume the die is fair, then, because the sum of the probabilities of the outcomes must equal 1, the probability of any particular outcome must equal , using the classical method. We write

For the experiment in Example 4, find the following probabilities:

(The solutions are shown in Appendix A.)

EXAMPLE 5 List all outcomes in a sample space using a tree diagram

Suppose our experiment is to toss a fair coin twice.

Solution

Suppose our experiment is to toss a fair coin three times.

(The solutions are shown in Appendix A.)

EXAMPLE 6 Classical method: Tossing a fair coin twice

Find the probability of obtaining one heads and one tails when a fair coin is tossed twice.

Solution

It is reasonable to assume that the outcomes in the sample space {HH, HT, TH, TT} are equally likely. The coin doesn’t remember what occurred at Stage 1, so the probabilities at Stage 2 are precisely the same as at Stage 1. Also, recall from the Law of Total Probability that the sum of the probabilities of all the outcomes in the sample space must equal 1. Thus, each of the four outcomes must have probability . Let be the event that one heads and one tails is obtained. Then , so . Thus,

For the experiment in Example 6, find the following probabilities:

(The solutions are shown in Appendix A.)

EXAMPLE 7 Finding probabilities for the experiment of tossing two fair dice

Imagine that you are playing Monopoly with your dormitory roommate, and the loser has to do the laundry for both of you for the rest of the semester. You have a hotel on Boardwalk, and if your roommate lands on it, you will surely win. Right now your roommate’s piece is on Short Line: if he or she rolls a 4, you will win and get your laundry done free for the remainder of the semester. Put into statistical terms, the experiment is to toss two fair dice and observe the sum of the two dice. Find the probability of rolling a sum of 4 when tossing two fair dice.

Solution

It is reasonable to assume that each of these outcomes in the sample space (Figure 3) is equally likely. The experiment of tossing two dice can be viewed as a two-stage experiment, where we add the result from the first die to the result from the second die. If a 5 appears on the first (say, dark green) die, and a 3 appears on the second (light green) die, the overall outcome is (5,3), with the resulting sum equal to 8. Note that the outcome (5,3) is not the same as the outcome (3,5), where the dark green die comes up 3 and the light green die comes up 5.

Let denote the event that your roommate rolls a sum equal to 4. Then the outcomes that belong in this event are , so . The outcomes are equally likely, so we can use the classical method for finding probabilities of events.

The probability that your roommate will land on Boardwalk on this throw of the dice is .

For the experiment in Example 7, find the following probabilities:

(The solutions are shown in Appendix A.)

EXAMPLE 8 Inappropriate use of the classical method

Watch a lot of YouTube? Hate the in-stream ads? According to Social Strand Media, 75% of in-stream ads are skippable (www.slideshare.net/SocialStrand/social-media-stats-2014). Suppose we choose one in-stream ad at random. Define the following events:

Determine whether the classical method can be used to assign probability to events and .

Solution

Because more than half of in-stream ads are skippable, if we choose one at random, we are more likely to find a skippable one than a non-skippable one. Therefore, the events and are not equally likely. It would be inappropriate to use the classical method of assigning probabilities for this experiment because the classical method can be used only when all the outcomes of an experiment are equally likely.

EXAMPLE 9 Simulating the long-term proportion of 6s in a fair die roll

Suppose we want to investigate the proportion of 6s we observe if we roll a fair die 100 times. We can use technology, such as the TI-83/84 used here, to help us simulate rolling a fair die a large number of times. A simulation uses methods such as rolling dice or computer generation of random numbers to generate results from an experiment. The actual die rolls from our simulation are shown here, in order, with the 6s in boldface.

Thus, the first die roll was a 1, so the proportion of 6s was . The second and third die rolls were 4s, so the proportion of 6s after 3 rolls was . On the fourth roll, a 6 appeared, so the proportion of 6s after the fourth roll was . Figure 4 provides a graph of the proportion of 6s in this simulation as the number of die rolls increased. Note that as the number of die rolls increases, the proportion of 6s tends to get closer to the horizontal line: .

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The simulation was rerun—this time with 1000 die rolls. The resulting graph of the proportion of 6s is provided in Figure 5. Note that as the number of die rolls increases, the proportion of 6s approaches the line , and the fit is tighter with 1000 die rolls than with 100. This is what we mean by “long-term proportion.”

Law of Large Numbers

As the number of times that an experiment is repeated increases, the relative frequency (proportion) of a particular outcome tends to approach the probability of the outcome.

Relative Frequency Method

If we can't use the classical method for assigning probabilities, then the Law of Large Numbers gives us a hint about how we can estimate the probability of an event. It often happens that previous information is available about the relative frequency of an event. Relative frequency information can be used to estimate the probability of the event.

Relative Frequency Method of Assigning Probabilities

The probability of event is approximately equal to the relative frequency of event . That is,

The relative frequency method is also known as the empirical method.

EXAMPLE 10 Relative frequency method

Using the experiment from Example 10, do the following:

(The solutions are shown in Appendix A.)

EXAMPLE 11 Probability models based on frequency tables

Table 2 contains the employment type for a sample of 1000 employed citizens of Fairfax County, Virginia.³ Use the data to construct the probability model by generating the relative frequencies and using the relative frequencies to estimate the probabilities for each employment type.

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Solution

We calculate the relative frequencies of each employment group by dividing the count (frequency) for each group by the sample size 1000. For example, the relative frequency for “Private Company” is . The relative frequency is then used to estimate the probability of selecting citizens who work at private companies in Fairfax County, Virginia. Filling in the remaining calculations produces the probability model in Table 3. Note that the table follows the Rules of Probability in that (a) each outcome has probability between 0 and 1, and (b) the sum of the probabilities of all the outcomes equals 1.0.

EXAMPLE 12 Random draws using a probability model

Suppose we consider the probabilities in Table 3 as population values. Use technology to simulate random draws using the probability model in Table 3.

Solution

Using the Step-by-Step Technology Guide on page 252, we drew samples of sizes 10, 100, 1,000, and 10,000 from the probability model in Table 3. The results are shown in Table 4.

Note that each relative frequency tends to approach its respective probability as the sample sizes grow larger.

Subjective probability refers to the assignment of a probability value to an outcome based on personal judgment.

EXAMPLE 13 Subjective method for assigning probability

A financial analyst is interested in the probability that the Dow Jones Industrial Average will go down today. Suppose the Chairman of the Federal Reserve warned against inflation in a major speech yesterday. How would the financial analyst use this information to subjectively assign the probability that the stock market will go down today?

Solution

Through long professional experience, the financial analyst has seen the Dow go down—often, the day after speeches by the Federal Reserve Chairman warning against inflation. She therefore assigns a subjective probability that the probability the Dow Jones Industrial Average will go down today is 75%. Although the exact subjective probability assigned by other financial analysts may differ somewhat, most will probably share the opinion that the Dow will decrease today.