Probability, Random Variables, and Probability Distributions

Twelve topics on quantifying uncertainty, starting from a table of counts and ending at the result the rest of the course rests on. Two-way tables and the three proportions they hold, simulation as a way to estimate a probability you cannot compute, the addition and multiplication rules and the two words students most often swap, mutually exclusive versus independent, random variables and how their means and variances combine, the binomial and normal models and when each one applies, and the Central Limit Theorem, which is why inference works at all.

Exam weight 15-25%12 topics

Topics

2.1

Tabular and Graphical Representations for the Distributions of Two Categorical Variables

Summary Statistics for Two Categorical Variables

Estimating Probabilities Using Simulation

Introduction to Probability

Mutually Exclusive Events

Conditional Probability

Independent Events and Unions of Events

Introduction to Random Variables and Probability Distributions

Parameters of Random Variables

The Binomial Distribution

The Normal Distribution

Sampling Distributions and the Central Limit Theorem

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Legal probability

0 ≤ P(A) ≤ 1, and all outcomes of the sample space sum to 1

Complement

P(not A) = 1 − P(A). The route to every "at least one" question

Addition rule

P(A or B) = P(A) + P(B) − P(A and B). Subtract the overlap or you count it twice

Mutually exclusive

P(A and B) = 0, so P(A or B) = P(A) + P(B). NOT the same as independent

Conditional

P(A | B) = P(A and B) / P(B). The condition sets the DENOMINATOR

P(A|B) vs P(B|A)

different questions, different denominators, usually different answers

Multiplication rule

P(A and B) = P(A) · P(B | A). It reduces to P(A)P(B) only when independent

Independence test

P(A | B) = P(A), or equivalently P(A and B) = P(A)P(B)

Long run, not short run

a probability describes many repetitions; a streak owes the next trial nothing

Expected value

E(X) = Σ x · P(x). A long-run average, usually not a possible outcome

Variance of X

Var(X) = Σ (x − μ)² · P(x), and SD is its square root

Shift and scale

E(aX + b) = a·E(X) + b, SD(aX + b) = |a|·SD(X): adding b never changes spread

Sums and differences

E(X ± Y) = E(X) ± E(Y). For INDEPENDENT X and Y, Var(X ± Y) = Var(X) + Var(Y): variances ADD even for a difference

Combining spread

add VARIANCES, never standard deviations

Binomial setting

fixed n, two outcomes, constant p, independent trials. Check all four

Binomial probability

P(X = k) = nCk · p^k · (1 − p)^(n − k). The counting coefficient is not optional

Binomial parameters

μ = np, σ = √(np(1 − p))

z-score

z = (x − μ) / σ: how many SDs from the mean, and in which direction

Empirical rule

about 68 / 95 / 99.7 percent within 1 / 2 / 3 SDs, for a NORMAL distribution only

Sampling distribution of x̄

mean μ, standard deviation σ / √n. The spread of the STATISTIC, not of the data

Central Limit Theorem

for large enough n the distribution of x̄ is approximately normal, whatever the population's shape

Unit 2 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 12 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 1 through 2, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 42-question section, your choice. Even coverage means this is a retention check rather than a score estimate.

20 or 42questions

25topics

29codes covered

Begin assessment →