This unit starts with symbols. A sentence such as “students who do drama or band but not both” becomes an expression that can be tested, shaded and proved equal to another. Counting three overlapping sets then needs a rule, and a proof that every region is counted once.
Probability is built next as a small system: three accepted statements, from which the complement rule and the addition rule are proved. Independence gives a calculation that tests whether one event changes the chance of another.
The last two lessons are about evidence. Conditional probability is read from tables, Venn diagrams and trees, and then reversed, which is what a screening test needs. The unit ends with a decision that must be justified with numbers.
The lessons, in order
- Lesson 1 · Set laws and De Morgan's laws Set expressions for two and three sets, De Morgan's laws, the distributive laws, shading regions, and proving two set expressions equal by following an element. Pick one of nine set expressions on a three-set Venn diagram, see which regions are shaded and which other expressions shade the same ones. No calculator
- Lesson 2 · Counting a union of three sets The rule for n(A ∪ B ∪ C), why every region is counted once, using it backwards, counting elements in exactly two sets, and the pattern for four sets. Choose a survey and a step, and see how many times each region of a three-set Venn diagram has been counted so far. No calculator
- Lesson 3 · Axioms and proving probability rules The three axioms, the words axiom, theorem and corollary, proofs of P(A′) = 1 − P(A), P(∅) = 0 and the addition rule, and the limits on P(A ∩ B). Set P(A), P(B) and P(A ∩ B) as shares of one whole and see the four regions, the union and whether such a model can exist. No calculator
- Lesson 4 · Independent events The test P(A ∩ B) = P(A) × P(B), finding unknowns for independent events, independent versus mutually exclusive, and a proof that complements of independent events are independent. Set P(A), P(B) and the overlap, and compare the overlap with the share chance would give. No calculator
- Lesson 5 · Conditional probability P(B | A) as P(A ∩ B) ÷ P(A), read from two-way tables, Venn diagrams and trees (with and without replacement), the multiplication rule, and P(A | B) against P(B | A). Choose one of four conditional questions on a two-way table and see the group that is given and the share inside it. No calculator
- Lesson 6 · Reversing a conditional probability: screening tests P(A | B) from P(B | A) with a tree and with natural frequencies, the total probability of a positive result, false positives, and justifying a screening decision. Set how common a condition is and how the test behaves, and see how many of those who test positive really have it. No calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.