### What is an event?

Probability is used to measure the likelihood of something happening. Implicit in the idea of likelihood is chance. We are uncertain what will happen. An experiment is a process that generates uncertain occurrences. These occurrences are called the outcomes of the experiment.

For instance, suppose a manufacturer is producing batteries that are sold in a two pack. If a package of batteries is selected from the production line, the batteries in the package may be examined to determine whether they work or are defective. The process of examining whether the batteries in the package are defective is an experiment. The outcome of the experiment may be listed by indicating whether each battery is working (W) or defective (D).

We can specify the first outcome of the experiment as (W, W). Other outcomes can be written in a similar manner. Written this way, this first letter indicates whether the first battery in the package is working or defective. The second letter indicates whether the second battery in the package is working or defective. We can refer to these outcomes collectively as

The experiment is carried out many times with each outcome being uncertain. These repetitions of the experiment are called trials.

The letter *S* is used to denote the sample space. The outcomes in the sample space are usually enclosed in brackets.

### Example 1 Find the Sample Space

If the battery producer examines the two-pack of batteries and notes the number of defective batteries, find the sample space.

**Solution** A two-pack of batteries may have 0, 1, or 2 defective batteries in it. Since the sample space is the set of all possible outcomes,

A tree diagram is useful for listing all of the outcomes from an experiment. For the battery packaging, we draw a pair of line segments from a common starting point to indicate whether the first battery works or does not work.

These line segments form the first branches of the tree. From each of these possibilities, another branch is drawn to indicate what might happen when the second battery in the package is examined.

By examining the tree diagram from left to right, we can list all of the outcomes in the sample space.

A similar strategy can be used for experiments that lead to more complicated branching.

### Example 2 Find the Sample Space

A marketing company wishes to survey a group of cell phone customers regarding their phone usage. On the first question of the survey, they will ask whether the customer uses a smartphone. On the second question, they ask whether they are on a family share plan. On a third question in the survey, they ask whether the customer has a texting plan. The answers to the questions are recorded as yes (Y) or no (N). If you consider the administration of the survey to be an experiment, find the sample space of the experiment.

**Solution** Construct a tree diagram like the one shown below.

If we correspond letters to these branches we can write the sample space as

Each outcome corresponds to an ordered triple. This is similar to the ordered pairs we often graph in algebra but with the entries inside the parentheses matching the answer to the questions. Since the survey has three questions on it, we need three sets of branches to specify all possible outcomes and three entries inside of the parentheses.

Often we are interested in a portion of the sample space. An event is any collection of outcomes from an experiment. We represent events with capital letters.

### Example 3 Find the Event

The marketing company is interested in several different events. Specify the outcomes that make up each of the events below.

a. The event *A*, all three questions are answered yes.

**Solution** From the tree diagram above, we found the sample space

The event *A* corresponds to the outcome where each entry in the ordered triple is *Y*,

b. The event* B*, two of the questions are answered no.

**Solution** We need to find all of the outcomes in the sample space that contain 2 *N*’s. This event is

c. The event *C*, the last question is answered yes.

**Solution** We need to find all outcomes where the last question was answered *Y*. This event is

In each of the parts above, the order in which the outcomes are listed is irrelevant. In other words, we could also write a collection like *B* as {(*N, Y, N*), (*Y, N, N*), (*N, N, Y*)} or {(*Y, N, N*), (*N, N, Y*), (*N, Y, N*)} . As long as the outcomes are listed inside the brackets, the event is the same. Similarly, the outcomes in the sample space may be listed in any order.

### Example 4 Find the Event

Breweries are classified by the amount of beer they produce in a year. The American Brewers Association defines a microbrewery as a brewery that produces less than 15,000 barrels per year. A nanobrewery is a very small brewery where beer is produced in very small batches.

One nanobrewery serves only three beers at a time in its tasting room. The owner conducts an experiment where he keeps track of the first two beers each customer purchases. He uses the letter *b* to indicate brown ale, *p* for pale ale, and *l* for lager, and *n* for no beer ordered. For instance, *bn* indicates that the first beer ordered is a brown ale and no second beer was ordered. List the outcomes in each of the events listed below.

a. The event *A*, only one beer is ordered.

**Solution** For this event, the second letter must be *n*. If we represent the event with the letter *A*,

b. The event *B*, the same beer is ordered for the first and second beer.

**Solution** List out all of the outcomes where the letters representing a beer match,

Note that *nn* is not listed since the event assumes a beer is ordered.