AP®︎ Statistics (pre-2026): Unit 4 Flashcards
Master key terms and definitions for Unit 4 of AP Statistics (pre-2026) – Probability, Random Variables, and Probability Distributions to help you prep for quizzes and the AP exam.
Patterns in Data Do Not Necessarily Mean Non-Randomness
An apparent trend or difference can occur by chance and does not alone prove a real effect.
An apparent trend or difference can occur by chance and does not alone prove a real effect.
Possible follow-up questions about whether an observed relationship is due to chance or a systematic cause.
Differences in data caused by chance alone, without a systematic pattern or cause.
Differences in data caused by a consistent influence rather than chance alone.
Unpredictable error causes scatter, while consistent error pushes results in a particular direction.
A chance-driven process whose individual results cannot be predicted with certainty.
One repetition of a random process that produces a result.
The single result produced by one trial of a random process.
A model of a random process that uses chance to imitate real-world outcomes.
A randomizing tool, such as digits, coins, or dice, used to model outcomes.
The actual long-run probability of an outcome or event in the real process.
As trials increase, simulated proportions tend to approach the true probability.
Assign chance outcomes, define one trial, repeat many times, record results, and estimate probability.
The proportion of trials with an outcome or event, used to estimate probability.
A description of a chance process listing all outcomes and their probabilities.
The set of all possible non-overlapping outcomes of a random process.
When outcomes are equally likely, probability equals favorable outcomes divided by total outcomes.
Probabilities are between 0 and 1 inclusive; 0 is impossible and 1 is certain.
The probabilities of all outcomes in the sample space add up to 1.
In repeated trials, probability is the proportion of times an event occurs in the long run.
State probability as a chance or long-run proportion using the situation's actual variables and outcomes.
All outcomes not in an event, written as E' or E^c, with probability 1 - P(E).
A set of one or more outcomes from a random process or sample space.
For disjoint events, P(A ∪ B) = P(A) + P(B) because there is no overlap.
Disjoint events cannot occur together; independent events do not affect each other's probabilities.
Events with no outcomes in common, so their intersection has probability 0.
The probability that both events occur, written as P(A ∩ B) or P(A and B).
The probability of both events occurring equals P(A ∩ B) = P(A) · P(B|A).
A cue word indicating a conditional probability, meaning the sample space is restricted to the stated condition.
A table of counts or proportions for two categorical variables, useful for finding joint and conditional probabilities.
A set diagram using overlapping regions to show events, intersections, and conditional relationships.
A branching diagram for multistage chance processes, where branch probabilities are often conditional probabilities.
The probability of event A given event B is P(A|B)=P(A∩B)/P(B).
Two events where knowing one occurred does not change the other's probability.
If events are independent, then P(A|B)=P(A), P(B|A)=P(B), and P(A ∩ B)=P(A)P(B).
Add individual probabilities and subtract overlap: P(A ∪ B)=P(A)+P(B)-P(A ∩ B).
The probability that event A or event B or both occurs, written as P(A ∪ B).
A numerical outcome whose value depends on the result of a chance process.
Discrete has countable values; continuous can take any value in an interval.
Probabilities are between 0 and 1, and all probabilities add to 1.
A table or function giving the probability that the variable is less than or equal to each value.
Use expressions like P(X = n) or P(X ≤ n) to name event probabilities.
Add probabilities for all values that satisfy the condition, including endpoints when stated.
Describe the distribution’s shape, center, and spread in context of the population.
A table, graph, or function that gives the probability of each possible value of a random variable.
A fixed numerical value describing a population characteristic or a random variable's distribution.
The probability-weighted average of squared deviations from the mean.
State the average and typical spread using the variable's units and the population context.
The long-run average, found by summing each value multiplied by its probability.
The square root of the probability-weighted squared deviations from the mean.
Changing a variable by adding a constant, multiplying by a constant, or both.
Mean becomes a + bμx, standard deviation becomes |b|σx, and shape stays the same if b > 0.
An expression formed by multiplying variables by constants and then adding them, like aX + bY.
For aX + bY, the mean is aμx + bμy.
Two variables where knowing one value does not change the other's probability distribution.
Add means for sums and subtract means for differences, keeping the order of subtraction.
Adding shifts center only, while multiplying changes center and spread and may reflect shape.
For independent variables, variances add in sums or differences, then take the square root for standard deviation.
Use known probability rules exactly, or estimate probabilities from many simulated trials with random numbers.
The number of ways to select x successes from n trials, written as a combination.
PDF gives one exact value; CDF gives the probability up to and including a value.
Simulate many sets of n trials, count successes each time, and use relative frequencies as estimates.
A discrete variable counting successes in n independent trials with constant success probability p.
For a binomial random variable, the probability of exactly x successes in n independent trials is P(X = x) = (n choose x)p^x(1 − p)^(n − x).
A binomial setting has a fixed number of trials, independent trials, two outcomes, and a constant probability of success.
For X counting successes in n trials, μx = np.
For X counting successes in n trials, σx = √(np(1 - p)).
State mean or standard deviation in context, with units, for the population or situation described.
When sampling without replacement, independence is reasonable if n < 0.10N.
A discrete variable counting the trial number on which the first success occurs.
Independent trials, two outcomes, constant success probability p, and interest in the first success.
P(X = x) = (1 - p)^(x - 1)p for x = 1, 2, 3, ....
The expected number of trials until the first success, equal to 1/p.
The spread in trials until the first success, equal to sqrt(1 - p)/p.
Positive integers 1, 2, 3, ... because the first success can occur on any trial.
Binomial counts successes in n fixed trials; geometric counts trials until the first success.
Describe mean or standard deviation as numbers of trials until the first success in context.
Always skewed right, with larger trial numbers becoming progressively less likely.
