Practice problems for these concepts can be found at:

- Probability and Random Variables Multiple Choice Practice Problems for AP Statistics
- Probability and Random Variable Free Response Practice Problems for AP Statistics
- Probability and Random Variables Cumulative Review Problems for AP Statistics
- Probability and Random Variables Rapid Review for AP Statistics

The second major part of a course in statistics involves making *inferences* about populations based on sample data (the first was *exploratory data analysis*). The ability to do this is based on being able to make statements such as, "The probability of getting a finding as different, or more different, from expected as we got by chance alone, under the assumption that the null hypothesis is true, is 0.6." To make sense of this statement, you need to have a understanding of what is meant by the term "probability" as well as an understanding of some of the basics of probability theory.

An **experiment or chance experiment (random phenomenon)**: An activity whose outcome we can observe or measure but we do not know how it will turn out on any single trial. Note that this is a somewhat different meaning of the word "experiment" than we developed in the last chapter.

**example:**if we roll a die, we know that we will get a 1, 2, 3, 4, 5, or 6, but we don't know

*which*one of these we will get on the next trial. Assuming a fair die, however, we

*do*have a good idea of approximately what proportion of each possible outcome we will get over a large number of trials.

**Outcome:** One of the possible results of an experiment (random phenomenon).

**example:**the possible outcomes for the roll of a single die are 1, 2, 3, 4, 5, 6. Individual outcomes are sometimes called

**simple events**.

### Sample Spaces and Events

**Sample space:** The set of all possible outcomes, or simple events, of an experiment.

example:For the roll of a single die, S = {1, 2, 3, 4, 5, 6}.

**Event:** A collection of outcomes or simple events. That is, an event is a subset of the sample space.

example:For the roll of a single die, the sample space (all outcomes or simple events) is S = {1, 2, 3, 4, 5, 6}. Let event A = "the value of the die is 6." Then A = {6}. Let B = "the face value is less than 4." Then B = {1, 2, 3}. Events A and B are subsets of the sample space.

example:Consider the experiment of flipping two coins and noting whether each coin lands heads or tails. The sample space is S = {HH, HT, TH, TT}. Let event B = "at least one coin shows a head." Then B = {HH, HT, TH}. Event B is a subset of the sample space S.

**Probability of an event**: the relative frequency of the outcome. That is, it is the fraction of time that the outcome would occur if the experiment were repeated indefinitely. If we let E = the event in question, *s* = the number of ways an outcome can succeed, and *f* = the number of ways an outcome can fail, then

Note that *s* + *f* equals the number of outcomes in the sample space. Another way to think of this is that the probability of an event is the sum of the probabilities of all outcomes that make up the event.

For any event A, *P*(A) ranges from 0 to 1, inclusive. That is, 0 ≤ *P*(A) ≤ 1. This is an algebraic result from the definition of probability when success is guaranteed (*f* = 0, *s* = 1) or failure is guaranteed (*f* = 1, *s* = 0).

The sum of the probabilities of all possible outcomes in a sample space is one. That is, if the sample space is composed of *n* possible outcomes,

**example**: In the experiment of flipping two coins, let the event A = obtain at least one head. The sample space contains four elements ({HH, HT, TH, TT}). *s* = 3 because there are three ways for our outcome to be considered a success ({HH, HT, TH}) and *f* = 1.

Thus

**example:** Consider rolling two fair dice and noting their sum. A sample space for this event can be given in table form as follows:

Let B = "the sum of the two dice is greater than 4." There are 36 outcomes in the samples space, 30 of which are greater than 4. Thus,

Furthermore,

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