Probability & Random Variables

What this unit covers

The topics below follow the published Statistics course framework for Unit 4. This unit is worth 10–20% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Probability Rules14 min · 3 objectivesApply the complement, addition, and multiplication rules · Distinguish mutually exclusive events from independent events · Use the general addition rule to avoid double-counting
• Random Variables: Mean & Variance14 min · 3 objectivesCompute the expected value (mean) of a discrete random variable · Compute and interpret the variance and standard deviation of a random variable · Apply rules for the mean and variance of transformed and combined variables
• The Binomial Distribution15 min · 3 objectivesVerify the four conditions (BINS) for a binomial setting · Compute binomial probabilities and the mean and standard deviation · Distinguish "exactly," "at most," and "at least" probability requests
• The Geometric Distribution13 min · 3 objectivesRecognize a geometric setting and contrast it with a binomial setting · Compute the probability that the first success occurs on trial k · Find the expected number of trials until the first success

Formulas in Unit 4

Every term in Unit 4

All 20 terms we publish for Probability & Random Variables, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Combining random variables

Means always add: μ(X ± Y) = μX ± μY. Variances add only when X and Y are independent, and they add even for a difference.

Law of large numbers

As the number of trials grows, the observed proportion approaches the true probability. It says nothing about short runs.

Complement rule

P(not A) = 1 − P(A). Often the fastest route to "at least one" problems.

Addition rule

P(A or B) = P(A) + P(B) − P(A and B). The subtraction avoids double-counting the overlap.

Mutually exclusive vs independent

Mutually exclusive events cannot both occur; independent events do not affect each other's probability. Mutually exclusive events with non-zero probability are never independent.

Conditional probability

P(A | B) = P(A and B)/P(B). The condition restricts the sample space to B.

Checking independence

A and B are independent when P(A | B) = P(A). Verify numerically rather than assuming.

Expected value

E(X) = Σx·P(x), the long-run average over many repetitions. It need not be an attainable value.

Variance of a random variable

Var(X) = Σ(x − μ)²P(x). Standard deviation is its square root, interpreted as typical distance from the expected value.

Binomial setting

Binary outcomes, Independent trials, fixed Number of trials, and constant probability of Success — BINS.

Binomial mean and standard deviation

μ = np and σ = √(np(1 − p)).

Geometric setting

Same as binomial but counting trials until the first success, so the number of trials is not fixed. Mean is 1/p.

10% condition

When sampling without replacement, trials are close enough to independent provided the sample is no more than 10% of the population.

Simulation design

State the model, how one trial is run, what counts as a success, and how many trials. Then report the estimated probability as a proportion.

"At least one" problems

Compute the complement — the probability of none — and subtract from 1. Far faster than summing every case.

Two-way table probabilities

Joint probability uses the grand total as the denominator; conditional probability uses the row or column total.

Tree diagrams

Multiply along a branch for a joint probability, then add across branches that satisfy the condition.

Binomial vs geometric on the exam

A fixed number of trials counting successes is binomial; counting trials until the first success is geometric.

Linear transformation of a random variable

For Y = a + bX, the mean transforms the same way but the standard deviation is multiplied by |b| and the addition has no effect on spread.

Independence of two random variables

Required before adding variances. Without it, the variance of a sum is not the sum of the variances.

What examiners penalize here

• When combining independent random variables, **variances add even when you subtract the variables**. To find the standard deviation of a difference X − Y, compute σ²_X + σ²_Y first, then take the square root. Never subtract standard deviations, and never add standard deviations directly.
• The quickest way to tell binomial from geometric: is the *number of trials fixed*? Fixed n, count successes → **binomial**. Keep going until the first success, count the trials → **geometric**. The phrase "until" almost always means geometric.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 9 units of AP Statistics

1. Unit 1 · Exploring One-Variable Data
2. Unit 2 · Exploring Two-Variable Data
3. Unit 3 · Collecting Data
4. Unit 4 · Probability & Random Variables
5. Unit 5 · Sampling Distributions
6. Unit 6 · Inference for Proportions
7. Unit 7 · Inference for Means
8. Unit 8 · Inference for Categorical Data: Chi-Square
9. Unit 9 · Inference for Quantitative Data: Slopes