Probability, Random Variables, and Probability Distributions
Use two-way tables to compare groups, learn how probability rules fit real situations and build models for random outcomes. Finish by separating individual observations from statistics that vary across repeated samples.
Before you start: Review variable types, proportions and quantitative distributions from Unit 1. Be comfortable with fractions, percentages, square roots and reading a table.
Explore all 12 topics
Choose a topic to open its study notes, worked examples, visuals and practice. Follow the cards in order, or jump to the skill you want to review.
Compare two categorical variables
Use two-way tables, displays and conditional proportions to describe association.
- Topic 2.1
Tabular and Graphical Representations for Two Categorical Variables
Read two-way tables and categorical displays. Compare groups with conditional proportions rather than unequal raw counts.
Study Topic 2.1 - Topic 2.2
Summary Statistics for Two Categorical Variables
Calculate joint, marginal and conditional relative frequencies. Use group distributions to describe evidence of association.
Study Topic 2.2
Reason with probability
Build simulations and select rules for complements, intersections, conditions and unions.
- Topic 2.3
Estimating Probabilities Using Simulation
Specify a random model, define one trial and repeat it. Estimate an event probability from the simulation results.
Study Topic 2.3 - Topic 2.4
Introduction to Probability
Define the sample space and events. Calculate simple probabilities and complements, and interpret long-run relative frequency.
Study Topic 2.4 - Topic 2.5
Mutually Exclusive Events
Recognize when two events cannot occur together. Use the intersection to justify that events are mutually exclusive.
Study Topic 2.5 - Topic 2.6
Conditional Probability
Restrict the sample space to the given event. Calculate conditional probability and apply the general multiplication rule.
Study Topic 2.6 - Topic 2.7
Independent Events and Unions of Events
Check independence using probabilities. Calculate a union without counting the intersection twice.
Study Topic 2.7
Model random variables
Define distributions, interpret parameters and recognize a binomial count.
- Topic 2.8
Introduction to Random Variables and Probability Distributions
Define a random variable and its possible values. Construct a valid discrete probability distribution and cumulative probabilities.
Study Topic 2.8 - Topic 2.9
Parameters of Random Variables
Calculate the mean, variance and standard deviation of a random variable. Explain these values using the situation and units.
Study Topic 2.9 - Topic 2.10
The Binomial Distribution
Check the binomial setting, then calculate count probabilities, mean and standard deviation. Distinguish exactly, at most and at least.
Study Topic 2.10
Use normal and sampling models
Find normal areas and cutoffs, then distinguish individuals from repeated-sample statistics.
- Topic 2.11
The Normal Distribution
Recognize a normal model, standardize values and find areas or cutoffs. Match a shaded region to the probability question.
Study Topic 2.11 - Topic 2.12
Sampling Distributions and the Central Limit Theorem
Distinguish population, sample and sampling distributions. Explain sample-to-sample variation and what the central limit theorem describes.
Study Topic 2.12
The big picture
A school records travel method and whether each student arrives on time. A two-way table describes this sample. Probability can model a future arrival outcome, and a random variable can count on-time arrivals over several days. Each step asks a different question.
Keep track of what one outcome means. A person, an event, a count over several trials and a statistic from a repeated sample are different objects, even when the same situation is used to describe them.
By the end of this unit, you should be able to:
- Calculate joint, marginal and conditional proportions from a two-way table.
- Explain mutually exclusive events, independent events, intersections and unions.
- Design a simulation whose random outcomes match the situation.
- Choose and use discrete, binomial and normal probability models appropriately.
- Interpret a random variable’s mean and standard deviation in context.
- Distinguish population, sample and sampling distributions and explain the role of sample size.
Your study roadmap
Use these stages as a learning order. If you are revising, choose the stage that matches the skill you need to practise. Each stage links back to its topic cards above.
- Stage 01 · Topics 2.1–2.2
Compare two categorical variables
Use two-way tables, displays and conditional proportions to describe association.
- Stage 02 · Topics 2.3–2.7
Reason with probability
Build simulations and select rules for complements, intersections, conditions and unions.
- Stage 03 · Topics 2.8–2.10
Model random variables
Define distributions, interpret parameters and recognize a binomial count.
- Stage 04 · Topics 2.11–2.12
Use normal and sampling models
Find normal areas and cutoffs, then distinguish individuals from repeated-sample statistics.
For each lesson: read the key idea, work through an example, try practice without the solution, then compare your reasoning with the explanation. Revisit the step you missed before moving on.
Revise with a purpose
Choose the route that fits your goal. While revising, practise explaining the method and result; a correct number on its own may leave the question unanswered.
Learning for the first time
Study the two-way-table stage before probability notation. Work through probability rules, then discrete distributions and binomial counts. Finish with normal models and repeated sampling. Open the starting lesson →
Revising probability wording
Use Topics 2.4–2.7. Translate “not,” “both,” “given” and “at least one” into the relevant event. Write the denominator or formula before entering calculator values. Open the starting lesson →
Revising model selection
Use Topics 2.8–2.12. State what X or the sample statistic represents, name the model, justify its conditions and identify exactly which values or tail area answer the question. Open the starting lesson →
Keep these distinctions clear
The denominator follows the condition
For P(A | B), the reference group is B. In a two-way table, restrict attention to that group before dividing. Changing the given event can change the probability.
Independent is different from mutually exclusive
Independence means learning that one event occurred does not change the probability of the other. Mutually exclusive events cannot happen together. Two disjoint events with positive probabilities are not independent.
A model needs a reason
A binomial variable counts successes in a fixed number of independent trials with the same success probability. A normal model describes a continuous variable under a justified distribution assumption. Choose from the setting, not a calculator menu.
Name the distribution you describe
A population distribution concerns individuals. A sample distribution concerns the observed sample values. A sampling distribution concerns a statistic from repeated samples of the same size.
Common mistakes to catch
- Using the grand total for every conditional probability
Restrict the denominator to the group specified by the given event. Review Topic 2.6
- Adding overlapping event probabilities without adjustment
Subtract the intersection once when calculating P(A ∪ B). Review Topic 2.7
- Using binomial commands for a variable that is not a binomial count
Check the fixed trial count, two outcome types, independence and constant success probability first. Review Topic 2.10
- Saying that the CLT makes the population normal
The theorem describes the distribution of sample means as sample size increases under appropriate assumptions. It does not change the distribution of individual values. Review Topic 2.12
Check your understanding
These are original, short retrieval questions with fictional teaching situations. Try each one before opening the answer. If your explanation is incomplete, follow the review link for the relevant topic.
Question 1
A fictional travel table has 60 bus students, of whom 40 arrive on time, and 40 car students, of whom 30 arrive on time. Find P(on time | bus) and P(on time | car).
Check answer 1
For bus students, use their group total: 40/60 ≈ 0.667. For car students, 30/40 = 0.75. The sample on-time proportions differ, giving descriptive evidence of association. These calculations alone do not establish a population difference or a causal effect of travel method.
Question 2
An event has probability 0.30. You use equally likely random digits 0–9, with 0, 1 and 2 representing success. One trial contains three digits: 2, 1, 7. How many successes occurred, and did “at least two successes” happen?
Check answer 2
There are two successes: 2 and 1. The digit 7 is a failure, so the event happened. The digit model gives success probability 3/10 = 0.30. To estimate the event probability, repeat the three-digit trial many times and divide the number of qualifying trials by the total number of trials.
Question 3
Events A and B are mutually exclusive, with P(A) = 0.30 and P(B) = 0.40. Are they independent?
Check answer 3
No. Their intersection probability is 0, but P(A)P(B) = 0.30(0.40) = 0.12. The probabilities do not satisfy the independence rule. If A occurs, B cannot occur, so the conditional probability of B changes.
Question 4
If P(A) = 0.40, P(B) = 0.30 and P(A ∩ B) = 0.10, find P(A ∪ B).
Check answer 4
P(A ∪ B) = 0.40 + 0.30 − 0.10 = 0.60. “A or B” includes outcomes in either event or both. Subtracting the intersection once corrects the double count.
Question 5
Let X count successes in 10 independent trials, each with success probability 0.20. Find its mean, standard deviation and probability of no successes.
Check answer 5
X is binomial with n = 10 and p = 0.20. Its mean is np = 2 successes; its standard deviation is √[np(1 − p)] = √1.6 ≈ 1.265 successes. P(X = 0) = 0.8¹⁰ ≈ 0.1074. The mean is a long-run average across repetitions of the ten-trial process, not a guarantee of two successes in every repetition.
Question 6
Individual travel times are right-skewed. Under suitable independent sampling assumptions, could the distribution of sample mean travel times become approximately normal for sufficiently large samples?
Check answer 6
Yes. The central limit theorem describes sample means across repeated samples. It does not say that individual travel times become normal. Keep the sample size fixed within a sampling-distribution description, and state which statistic is being repeated. How large is sufficient depends in part on the population’s shape.
Are you ready to move on?
Use this as a checklist: explain each item aloud or on paper without looking at the notes. A checked box is a reminder for your study session, not an assessment score.
If an item is not yet comfortable, choose the matching stage in the roadmap and retry that lesson’s practice. If these explanations are clear, work on mixed questions where the topic is not named for you.
Continue learning
Keep building your statistics skills
In Unit 3, probability and sampling models help you estimate population proportions and test claims using sample data.
Continue to Unit 3: Inference for Categorical Data: Proportions →
The unit sequence follows the AP Statistics course framework effective Fall 2026. Use this page to navigate NUM8ERS lessons and plan your revision.