Probability

Describe chance clearly, choose a justified model and calculate probabilities.

Twelve lessons cover sample spaces, data, Venn diagrams, independence, probability algebra and trees. Finish with multi-stage experiments and mixed practice; every lesson has learner-paced notes and worked solutions.

Sample spaces and probabilityList outcomes systematically, distinguish events from outcomes and calculate probabilities only when the equal-likelihood assumptions justify counting.
Probabilities from tables and histogramsChoose the correct population and denominator, calculate event probabilities from category tables and distinguish exact grouped counts from histogram estimates.
Two-event Venn diagramsBuild a two-event Venn diagram, interpret intersections and inclusive unions, and find only, neither and exactly-one probabilities without double counting.
Three-event Venn diagramsFill a three-set Venn diagram from the centre out, distinguish pairwise totals from pairwise-only regions, and calculate exactly-one and at-least-two probabilities.
Mutually exclusive eventsRecognise events that cannot occur together, use the correct addition rule, and distinguish disjoint, complementary, exhaustive and independent events.
Independent eventsTest independence using the intersection product, calculate probabilities with complements, and distinguish an assumed model from evidence in observed data.
Probability parameters and proofsFind valid parameter ranges, solve equations from independence and union information, reject impossible roots and prove identities using disjoint regions.
Tree diagrams with replacementRead successive probability branches, multiply along complete paths, add distinct paths for an event and use complements with independent repeated sampling.
Sampling without replacementUpdate counts after each draw, calculate dependent tree paths, compare with replacement and distinguish unchanged marginal probabilities from independence.
Conditional branches and total probabilityInterpret second-stage probabilities conditional on a route or group, form weighted totals across disjoint branches, and check what independence would require.
Multi-stage probabilityEnumerate three-stage events, distinguish exactly from at least, calculate first-success probabilities and state the independence needed when repeating whole experiments or sampling groups.
Probability: mixed practiceTest your choice of probability method with original sample-space, Venn, histogram, independence, parameter, tree and multi-stage problems with worked solutions.