1.1 — Sample Spaces & Counting Outcomes
Listing outcomes of probability experiments; tree diagrams and tables for one- and two-stage experiments.
1.2 — The Fundamental Counting Principle
Multiplicative principle for sequential independent choices; product rule \( n_1 \times n_2 \times \cdots \times n_k \).
1.3 — Mutually Exclusive Events & the Additive Principle
When events cannot occur together: \( P(A \cup B) = P(A) + P(B) \). Inclusive case: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
1.4 — Venn Diagrams & Set Operations
Union, intersection, complement, and difference of events; counting elements in unions of sets.
1.5 — Independent vs Dependent Events
Multiplicative principle for independent events; conditional probability for dependent events: \( P(A \cap B) = P(A) \cdot P(B \mid A) \).
2.1 — Factorial Notation
Definition \( n! = n(n-1)(n-2)\cdots 1 \); convention \( 0! = 1 \); evaluation and simplification of factorial expressions.
2.2 — Permutations of Distinct Objects
Number of arrangements of \( r \) objects from \( n \) distinct objects: \( P(n,r) = \dfrac{n!}{(n-r)!} \).
2.3 — Permutations with Identical Objects
Arrangements when some objects are indistinguishable: \( \dfrac{n!}{a!\,b!\,c!\cdots} \).
2.4 — Permutations with Restrictions
Cases where certain elements must (or must not) be adjacent, in specific positions, or treated as a single unit.
2.5 — Circular Permutations
Arrangements around a circle: \( (n-1)! \); applications to seating around tables and bracelet problems.
3.1 — Combinations: Counting Without Order
Number of subsets of size \( r \) from \( n \): \( C(n,r) = \binom{n}{r} = \dfrac{n!}{r!(n-r)!} \).
3.2 — Combination Identities
\( \binom{n}{r} = \binom{n}{n-r} \) and \( \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \); the symmetry property.
3.3 — Pascal's Triangle & Patterns
Constructing Pascal's triangle; recursion, row sums \( 2^n \), hockey-stick identity, Fibonacci numbers in diagonals.
3.4 — The Binomial Theorem
Expansion \( (a+b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \); finding specific terms.
3.5 — Combinations & Probability
Computing probabilities using combinations: \( P = \dfrac{\binom{\text{favourable}}{r}}{\binom{n}{r}} \).
4.1 — Theoretical & Empirical Probability
Theoretical \( P(A) = \dfrac{n(A)}{n(S)} \) for equally likely outcomes; empirical (relative-frequency) probability from data; subjective probability.
4.2 — Conditional Probability
\( P(B \mid A) = \dfrac{P(A \cap B)}{P(A)} \); restricting the sample space; Bayes-style reasoning.
4.3 — Discrete Random Variables & Probability Distributions
Definition of discrete random variable \( X \); probability distribution table; conditions \( 0 \le P(X=x) \le 1 \) and \( \sum P(X=x) = 1 \).
4.4 — Expected Value
\( E(X) = \displaystyle\sum_{i} x_i \cdot P(X = x_i) \); fair games; insurance pricing; long-run averages.
4.5 — Variance & Standard Deviation of Discrete Random Variables
\( \mathrm{Var}(X) = E(X^2) - [E(X)]^2 \); \( \sigma_X = \sqrt{\mathrm{Var}(X)} \).
5.1 — The Binomial Distribution
Fixed number of independent Bernoulli trials: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \). Mean \( np \), variance \( np(1-p) \).
5.2 — The Geometric Distribution
Number of independent Bernoulli trials until the first success: \( P(X = k) = (1-p)^{k-1} p \). Mean \( \tfrac{1}{p} \).
5.3 — The Hypergeometric Distribution
Sampling without replacement from a finite population: \( P(X=k) = \dfrac{\binom{r}{k}\binom{N-r}{n-k}}{\binom{N}{n}} \).
5.4 — Choosing the Right Distribution
Decision tree: replacement? fixed trials? counting trials or successes? — selecting binomial vs. geometric vs. hypergeometric.
6.1 — Continuous Random Variables & Density
Probability density vs. probability mass; area under a density curve; uniform continuous distribution.
6.2 — Properties of the Normal Distribution
Bell-shape, symmetry about \( \mu \), inflection points at \( \mu \pm \sigma \), total area = 1; the empirical (68-95-99.7) rule.
6.3 — Standardizing & z-Scores
\( z = \dfrac{x - \mu}{\sigma} \); using the standard-normal table to compute \( P(Z \le z) \).
6.4 — Normal Approximation to the Binomial
When \( np \ge 5 \) and \( n(1-p) \ge 5 \), \( X \approx N(np, \sqrt{np(1-p)}) \); continuity correction.
7.1 — Population, Sample & Sampling Methods
Census vs sample; simple random, stratified, systematic, cluster, voluntary-response, and convenience samples.
7.2 — Sources of Bias in Surveys
Sampling bias, non-response bias, response bias, measurement bias, household bias; identifying bias in real surveys.
7.3 — Measures of Central Tendency
Mean \( \bar{x} = \tfrac{\sum x_i}{n} \), median, mode; weighted mean; effect of outliers; trimmed mean.
7.4 — Measures of Spread
Range, IQR, variance \( s^2 = \dfrac{\sum(x_i-\bar{x})^2}{n-1} \), standard deviation \( s \); five-number summary, box-and-whisker plots.
7.5 — Frequency Distributions & Histograms
Class intervals, frequency, relative-frequency, cumulative-frequency tables; histograms and ogives; shape (symmetric, skewed).
8.1 — Scatter Plots & Correlation
Constructing scatter plots; describing the form, direction, and strength of a relationship; explanatory vs response variable.
8.2 — The Pearson Correlation Coefficient \( r \)
\( r = \dfrac{1}{n-1}\displaystyle\sum_{i=1}^{n} \dfrac{x_i-\bar{x}}{s_x}\cdot\dfrac{y_i-\bar{y}}{s_y} \); interpreting \( r \in [-1, 1] \); correlation \(\ne\) causation.
8.3 — Linear Regression & Line of Best Fit
Least-squares line \( \hat{y} = b_0 + b_1 x \) where \( b_1 = r\dfrac{s_y}{s_x} \); interpreting slope and intercept; using \( \hat{y} \) for prediction.
8.4 — Residuals & Goodness of Fit
Residual \( e_i = y_i - \hat{y}_i \); residual plots; identifying when a non-linear model fits better.
8.5 — Two-Variable Analysis & Causation
Cause-and-effect, common cause, reverse cause-and-effect, accidental and presumed relationships; lurking and confounding variables.
8.6 — Culminating Data-Management Investigation
Independent inquiry: develop a research question, collect/find primary or secondary data, apply two-variable analysis, present a written and oral report.
🎬Project guidelines
The Culminating Investigation contributes to the 30% Final Evaluation. See the Final Exam page for the rubric.
B1 — Classic Probability Paradoxes
Counter-intuitive probability problems that build deep conceptual understanding (Strand A enrichment).
B2 — Geometric Distribution & Normal Approximation
Fills explicit Strand B gaps: the geometric distribution (waiting-time discrete model) and the normal approximation to the binomial with continuity correction.
B3 — AP Statistics Overlap: Sampling Distributions & Hypothesis Testing
Bridge to AP Stats / first-year university: how sample statistics behave, the central limit theorem in action, and a clean introduction to hypothesis testing and the null hypothesis.
B4 — Statistical Literacy: Reading Polls & Margins of Error
Real-world skill of interpreting election polls, public-opinion studies, and margin-of-error claims in news media (Strand C / D enrichment).
B5 — Excel/Spreadsheet Workflow for the Culminating Investigation
Hands-on tooling: build a scatter plot, fit a least-squares trendline, display equation and R-squared. Exactly what students need for Strand D.