Starnes/Tabor, The Practice of Statistics for the AP® Course, 7e, Unit 4, English

random process

Generates outcomes that are determined purely

by chance.

probability

A number between 0 and 1 that describes the proportion of times an outcome of a random process would occur in a very long series of trials.

law of large numbers

If we observe more and more trials of any random process, the proportion of times that a specific outcome occurs approaches its probability.

simulation

Imitation of a random process in such a way that simulated outcomes are consistent with real-world outcomes.

probability model

Description of a random process that consists of two parts: a list of all possible outcomes and the probability of each outcome.

sample space

List of all possible outcomes of a random process.

event

A subset of the possible outcomes from the sample space of a random process. Events are usually designated by capital letters, like A, B, C, and so on.

complement rule

The probability that an event does not occur is 1 minus the probability that the event does occur. In symbols, P(Aᶜ) = 1 − P(A).

complement

The complement of event A, written as Aᶜ, is the event that A does not occur.

mutually exclusive

Two events A and B that have no outcomes in common and so can never occur together. That is, P(A and B) = 0.

addition rule for mutually exclusive events

If A and B are mutually exclusive events, P(A or B) = P(A) + P(B).

general addition rule

If A and B are any two events resulting from the same random process, then the probability that event A or event B (or both) occur is P(A or B) = P(A∪B) = P(A) + P(B) - P(A∩B).

Venn diagram

A figure that consists of one or more circles surrounded by a rectangle. Each circle represents an event. The region inside the rectangle represents the sample space of the random process.

intersection

The event "A and B" is called the intersection of events A and B. It consists of all outcomes that are common to both events, and is denoted by A∩B.

union

The event "A or B" is called the union of events A and B. It consists of all outcomes that are in event A, event B, or both, and is denoted by A∪B.

conditional probability

Probability that one event happens given that another event is already known to have happened. The probability that event A happens given that event B has happened is denoted by P(A|B).

independent events

Two events are independent if knowing whether or not one event has occurred does not change the probability that the other event will happen. In other words, events A and B are independent if P(A|B) = P(A|Bᶜ) = P(A) and P(B|A) = P(B|Aᶜ) = P(B).

general multiplication rule

For any random process, the probability that events A and B both occur can be found using the formula P(A and B) = P(A∩B) = P(A) * P(B|A).

tree diagram

A diagram that shows the sample space of a random process involving multiple stages. The probability of each outcome is shown on the corresponding branch of the tree. All probabilities after the first stage are conditional probabilities.

multiplication rule for independent events

If A and B are independent events, then the probability that A and B both occur is P(A and B) = P(A∩B) = P(A) * P(B).

random variable

Variable that takes numerical values that describe the outcomes of a random process.

probability distribution

Gives the possible values of a random variable and their probabilities. Gives the possible values of a random variable and their probabilities.

discrete random variable

Variable that takes a countable set of possible values with gaps between them on a number line. The probability of any event is the sum of the probabilities for the values of the variable that make up the event.

mean (expected value) of a discrete random variable

Describes the variable’s long-run average value over many, many trials of the same random process.

standard deviation of a discrete random variable

Measures how much the values of the random variable typically vary from the mean in many, many trials of the random process.

independent random variables

If knowing the value of X does not help us predict the value of Y, then X and Y are independent random variables. In other words, two random variables are independent if knowing the value of one variable does not change the probability distribution of the other variable.

binomial setting

Arises when we perform n independent trials of the same random process and count the number of times that a particular outcome (called a "success") occurs. The four conditions for a binomial setting are: Binary? The possible outcomes of each trial can be classified as "success" or "failure." Independent? Trials must be independent. That is, knowing the outcome of one trial must not tell us anything about the outcome of any other trial. Number? The number of trials n of the random process must be fixed in advance. Same probability? There is the same probability p of success on each trial.

binomial random variable

The count of successes X in a binomial setting. The possible values of X are 0, 1, 2, . . . , n.

binomial distribution

In a binomial setting, suppose we let X = the number of successes. The probability distribution of X is a binomial distribution with parameters n and p, where n is the number of trials of the random process and p is the probability of a success on each trial.

binomial coefficient

The number of ways to arrange x successes among n trials.

10% condition

When selecting a random sample of size n (without replacement) from a population of size N, we can treat individual observations as independent when performing calculations as long as n < 0.10N.

geometric setting

Arises when we perform independent trials of the same random process and record the number of trials it takes to get one success. On each trial, the probability p of success must be the same.

geometric random variable

The number of trials X that it takes to get a success in a geometric setting. The possible values of X are 1, 2, 3, . . . .

geometric distribution

In a geometric setting, suppose we let X = the number of trials it takes to get a success. The probability distribution of X is a geometric distribution with parameter p, the probability of a success on any trial.