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Unit 4: Probability, Random Variables, and Probability Distributions

12 dot points across 12 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do the mean and standard deviation change when we add, subtract, or rescale random variables?

Topic 4.9 Combining Random Variables: apply the rules for the mean and variance of a linear transformation and of sums and differences of random variables, adding variances (not standard deviations) for independent variables.
A focused answer to AP Statistics Topic 4.9, on transforming and combining random variables, how means and variances behave under scaling and addition, the add-the-variances rule for independence, and why variances add for differences too, with worked calculations.
10 min answer →

How does knowing one event has occurred change the probability of another?

Topic 4.5 Conditional Probability: calculate and interpret conditional probabilities using the definition and the multiplication rule, including from two-way tables and tree diagrams.
A focused answer to AP Statistics Topic 4.5, defining conditional probability, the multiplication rule, and computing conditional probabilities from two-way tables and tree diagrams, with full worked calculations.
10 min answer →

How can we estimate a probability by simulating a random process many times?

Topic 4.2 Estimating Probabilities Using Simulation: design and carry out a simulation using a chance device or random numbers to estimate a probability as a long-run relative frequency.
A focused answer to AP Statistics Topic 4.2, on designing and running simulations with random numbers to estimate probabilities, the four-step simulation method, and reading the estimate as a long-run relative frequency.
9 min answer →

What does it mean for two events to be independent, and how does that simplify the multiplication rule?

Topic 4.6 Independent Events and Unions of Events: determine whether events are independent, apply the multiplication rule for independent events, and combine the addition and multiplication rules to find probabilities of unions and intersections.
A focused answer to AP Statistics Topic 4.6, defining independence, the multiplication rule for independent events, the distinction from mutually exclusive, and combining rules for unions and intersections, with worked calculations.
10 min answer →

How do we tell whether a pattern we see is real or could easily have arisen by chance?

Topic 4.1 Introducing Statistics: Random and Non-Random Patterns? Recognize that random processes produce patterns, and that probability provides the framework for deciding whether an observed pattern is surprising or consistent with chance.
A focused answer to AP Statistics Topic 4.1, on why random processes still produce patterns, what randomness and short-run versus long-run behavior mean, and how probability frames whether an observed pattern is surprising.
9 min answer →

What are the basic rules every probability must obey, and how do we use the complement?

Topic 4.3 Introduction to Probability: apply the basic properties of probability (range, total of one, complement rule) and the law of large numbers to compute and interpret probabilities of events.
A focused answer to AP Statistics Topic 4.3, on the basic axioms of probability, the complement rule, sample spaces and equally likely outcomes, and the law of large numbers, with worked complement and basic probability calculations.
9 min answer →

How does a probability distribution describe all the possible values of a numerical random outcome?

Topic 4.7 Introduction to Random Variables and Probability Distributions: define discrete random variables, represent and interpret their probability distributions, and use them to find probabilities of events.
A focused answer to AP Statistics Topic 4.7, defining discrete random variables, the requirements of a valid probability distribution, cumulative probabilities, and interpreting distributions in context, with worked probability calculations.
9 min answer →

When does a setting follow a binomial distribution, and how do we compute its probabilities?

Topic 4.10 Introduction to the Binomial Distribution: identify binomial settings (BINS conditions) and use the binomial probability formula to find the probability of a given number of successes in a fixed number of trials.
A focused answer to AP Statistics Topic 4.10, on the binomial setting (the BINS conditions), the binomial probability formula, and computing exact and cumulative binomial probabilities, with full worked calculations.
10 min answer →

How do we find the long-run average and spread of a random variable from its distribution?

Topic 4.8 Mean and Standard Deviation of Random Variables: calculate and interpret the mean (expected value), variance, and standard deviation of a discrete random variable from its probability distribution.
A focused answer to AP Statistics Topic 4.8, on the expected value (mean), variance, and standard deviation of a discrete random variable, the weighted-average idea, and interpreting expected value as a long-run mean, with full worked calculations.
10 min answer →

When can we add probabilities directly, and what is the general addition rule?

Topic 4.4 Mutually Exclusive Events: identify mutually exclusive (disjoint) events and apply the addition rule, including the general addition rule that subtracts the overlap, to find the probability of a union.
A focused answer to AP Statistics Topic 4.4, defining mutually exclusive (disjoint) events, the addition rule for disjoint events, and the general addition rule that subtracts the intersection, with worked union calculations.
9 min answer →

What are the mean and standard deviation of a binomial distribution, and what shape does it take?

Topic 4.11 Parameters for a Binomial Distribution: calculate and interpret the mean and standard deviation of a binomial random variable using the shortcut formulas, and describe how the distribution's shape depends on n and p.
A focused answer to AP Statistics Topic 4.11, on the binomial mean np and standard deviation, why the shortcuts work, interpreting them in context, and how shape depends on n and p, with full worked calculations.
9 min answer →

How do we model the number of trials needed to get the first success?

Topic 4.12 The Geometric Distribution: identify a geometric setting (waiting for the first success), compute geometric probabilities, and find the mean of a geometric random variable.
A focused answer to AP Statistics Topic 4.12, on the geometric setting, the geometric probability formula, the mean of a geometric random variable, and how it differs from the binomial, with full worked calculations.
9 min answer →