This unit starts with sets and the symbols that name them: union, intersection and complement, drawn as Venn diagrams for two and three sets. Counting from a diagram shows why n(A ∪ B) = n(A) + n(B) − n(A ∩ B) and not a plain sum.
Then the same diagrams become probability. A sample space lists every equally likely outcome, and the probability of an event is a share of it. Tree diagrams follow the outcomes of two or three stages, with and without replacement. Two-way tables and Venn diagrams give probabilities within a group, and the last lesson asks what a probability really promises when the events are random.
A graphic display calculator (GDC) is optional all the way through. Lessons 4 and 6 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Union, intersection and complement The universal set U, union A ∪ B, intersection A ∩ B and complement A′, shading regions of a two-set Venn diagram, and De Morgan's laws. Slide through set expressions and see which of twelve numbers are in each, then compare two expressions region by region. No calculator
- Lesson 2 · Counting with Venn diagrams n(A ∪ B) = n(A) + n(B) − n(A ∩ B), finding a missing region of a two-set diagram, and three-set problems filled from the centre outwards. Change two totals and the overlap and watch the sum count the overlap twice, then fill a three-set diagram one layer at a time. No calculator
- Lesson 3 · Sample spaces and probability A sample space as a list or a two-way table, P(A) = n(A) ÷ n(S) for equally likely outcomes, the axioms 0 ≤ P(A) ≤ 1 and P(S) = 1, and P(A′) = 1 − P(A). Pick an event on a table of two dice and see how many of the 36 outcomes belong to it, then split 24 tickets into an event and its complement. No calculator
- Lesson 4 · Tree diagrams Two- and three-stage tree diagrams, multiplying along branches and adding routes, independent events and events without replacement, and P(at least one) = 1 − P(none). Build a tree for a bag of counters, press without replacement to see the second stage change, then see why at least one is 1 minus none. GDC optional
- Lesson 5 · Two-way tables and mutually exclusive events Probabilities from two-way tables and Venn diagrams, the probability within a given group, P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and mutually exclusive events. Read six probabilities from one table and see the group change the denominator, then slide the overlap of two events to zero. No calculator
- Lesson 6 · Randomness and experimental probability Randomness and runs, relative frequency as experimental probability, expected frequency P × n, how sample size affects reliability, and the difference between experimental and theoretical probability. See how long a run of the same face really is in 50 fair flips, then watch two runs of coin flips settle as the number of flips grows. GDC optional
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.