Probability does not predict the next outcome; it describes uncertainty under a stated model. This editable high school statistics lesson connects events and conditional probability to random variables, discrete distributions, expected value, binomial models, Bayes’ theorem, and decision analysis.
The packet includes ten worked examples, two checked diagrams, synthetic two-way data, derivations of the major formulas, forty sequenced exercises, and a separate fully worked teacher edition. It repeatedly asks students to name the relevant sample space, justify independence, distinguish a conditional from its reverse, and interpret an expected value without treating it as a guarantee.
Conditioning changes the sample space
For events $A$ and $B$ with $P(B)>0$,
$$ P(A\mid B)=\frac{P(A\cap B)}{P(B)}. $$
The phrase “given $B$” restricts attention to outcomes in $B$. This explains why $P(A\mid B)$ and $P(B\mid A)$ answer different questions.
The lesson uses a synthetic transportation table to compare
$$ P(\text{on time}\mid\text{bus})=\frac{60}{72} $$
with
$$ P(\text{bus}\mid\text{on time})=\frac{60}{85}. $$
It also distinguishes independence from mutual exclusivity. Independent events leave one another’s probabilities unchanged; nonempty mutually exclusive events make one another impossible.
Bayes’ theorem reverses a condition
The multiplication rule gives
$$ P(A\cap B)=P(B\mid A)P(A)=P(A\mid B)P(B). $$
Therefore
$$ P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}. $$
A frequency tree with $10{,}000$ synthetic cases shows how a low base rate can make a high-sensitivity result much less conclusive than students first expect. Counts, a two-way table, and the formula all produce the same posterior probability. The context is explicitly mathematical rather than medical advice.
A distribution assigns probability to numerical outcomes
A random variable maps outcomes to numbers. A valid discrete probability mass function satisfies
$$ P(X=x)\ge0 \quad\text{and}\quad \sum_xP(X=x)=1. $$
For the number of heads in two fair tosses, the values $0$, $1$, and $2$ have probabilities $1/4$, $1/2$, and $1/4$. The middle value is more likely because two ordered outcomes produce it.
The packet contrasts theoretical distributions derived from model assumptions with empirical distributions formed from observed relative frequencies.
Expected value is a long-run mean
For a discrete random variable,
$$ E(X)=\sum_x xP(X=x). $$
The expected value need not be a possible outcome. Variance and standard deviation describe spread around that mean:
$$ \operatorname{Var}(X)=E(X^2)-[E(X)]^2, \qquad \sigma=\sqrt{\operatorname{Var}(X)}. $$
The lesson proves $E(aX+b)=aE(X)+b$ from the weighted sum. It then uses games, warranties, payoffs, and costs to separate fairness, expectation, and risk.
The binomial formula comes from counting sequences
A binomial model requires a fixed number of independent trials, two outcomes per trial, and a constant success probability. For exactly $k$ successes in $n$ trials,
$$ P(X=k)=\binom nkp^k(1-p)^{n-k}. $$
The factor $p^k(1-p)^{n-k}$ is the probability of one specified success-failure order, while $\binom nk$ counts all such orders. Indicator variables then give the mean $np$; independence gives variance $np(1-p)$.
Students test whether scenarios really are binomial, use complements for “at least one,” construct a full distribution, and analyze multiple-choice guessing.
Decisions expose assumptions
Expected value can compare strategies, but it does not certify the probabilities, include omitted consequences, or decide a person’s tolerance for loss. The packet asks students to compare a guaranteed payoff with a variable one, analyze a raffle and a cost-risk choice, and challenge the assumptions in a weather-dependent planning decision.
Representative practice
The forty exercises are divided among:
- events, addition and multiplication rules, conditionals, and independence;
- theoretical and empirical discrete distributions;
- expectation, variance, standard deviation, fairness, and risk;
- binomial conditions, probabilities, means, and distributions;
- Bayes updates, source attribution, and expected-value decisions.
The teacher edition includes full calculations, proof checkpoints, diagnostic questions, model cautions, and scoring guidance.
Adapt it for a class
The student lesson and teacher edition are separate Markdown files. Teachers can use the conditional-probability section before a statistics unit, reserve Bayes’ theorem and expected-value decisions for an advanced extension, replace synthetic contexts, alter calculator expectations, or assign one practice strand at a time. School identity, pacing, standards language, numerical difficulty, and answer space remain editable.
Sources and boundaries
- Common Core State Standards for Mathematics, HSS-CP for conditional probability, independence, and probability rules.
- Common Core State Standards for Mathematics, HSS-MD for random variables, probability distributions, expected value, and decisions.
- OpenStax Introductory Statistics 2e, Chapters 3-4, CC BY 4.0, consulted for modern instructional scope.
- NIST/SEMATECH e-Handbook, Binomial Distribution for the probability mass function and model conditions.
All tables, screening counts, business cases, and decisions are synthetic teaching models. They are not medical, financial, insurance, engineering, or operational advice. Independent mathematics-teacher review and classroom trial remain outstanding.