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Probability, grade 11 (5 units)

How to work out a probability when an experiment has several stages, when something is already known, and when the same trial is repeated many times.Subjects

The course goes on from the probability of lower secondary school: equally likely outcomes, relative frequency, the outcome table and the tree diagram. The topics, in order:

  • events and the rules of probability: the sample space, simple and compound events, the union, the intersection and the complement in a Venn diagram, probability as a ratio of areas and as a relative frequency, the rules of probability;
  • conditional probability: the two-way table, 2×2\displaystyle 2\times2 and 3×2\displaystyle 3\times2, the conditional probability P(A/B)\displaystyle P(A/B), dependent and independent events;
  • multi-stage experiments: the probability tree, branches that depend on the path, choosing without putting back, the law of total probability and the reverse question – what happened at the first stage;
  • Bernoulli’s formula: the multiplication principle, factorials and binomial coefficients, the probability of k\displaystyle k successes in n\displaystyle n trials, “at least” and “at most”.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The sample space and eventsLearningThe sample space, simple and compound events, the certain and the impossible event, the probability of equally likely outcomes, and a sample space whose outcomes are not equally likely.
  2. Operations on events and the Venn diagramLearningThe union, the intersection and the complement, disjoint events, reading a Venn diagram – outcomes and numbers in each of its parts – and "neither A nor B".
  3. The rules of probabilityLearningThe rules of probability – bounds, an event inside another, the complement, the union and disjoint events – probability as a ratio of areas, and probability as the relative frequency of many repeats of an experiment.
  4. Practice: events and the rules of probabilityPracticeEvents in the outcome table, the Venn diagram of a survey, the probability of a union and of a complement, disjoint events, a ratio of areas and a relative frequency.
  5. The two-way tableLearningThe two-way table, 2×2 and 3×2 – numbers and probabilities, the intersection, the union and the complement from the table, completing a table from percentages, and a table with an unknown.
  6. Conditional probabilityLearningThe probability of an event when another event is known to have happened – from the two-way table and by the formula, a table built from conditional percentages and the area model, the difference between P(A/B) and P(B/A), and what the value of P(A/B) says about the two events.
  7. Dependent and independent eventsLearningEvents where knowing that one happened does not change the probability of the other, three equivalent tests of independence, a test in a two-way table, disjoint events that are dependent, and working out probabilities of independent events – with an unknown too.
  8. Practice: conditional probability and independencePracticeA 3×2 two-way table, a table from conditional percentages in the area model, a conditional probability with an unknown, and independent events – calculations, an equation and a short proof.
  9. The probability tree of a multi-stage experimentLearningStages, branches and paths, the probability of a path as a product and why, an event of several paths, choosing without putting back, independent stages, an experiment of three stages and "at least one" by the complement.
  10. The law of total probability and the reverse questionLearningThe probability of an outcome at the second stage as the sum of its paths – in a tree and in the area model, the reverse question: what happened at the first stage when the second is known, a medical test for a rare disease and the base rate in a population.
  11. Practice: multi-stage experimentsPracticeAn entrance exam – three groups and an unknown, digits chosen without putting back, a match of three sets, guessing in multiple-choice questions, and a reverse question in an experiment of three stages.
  12. The multiplication principle, factorials and binomial coefficientsLearningThe multiplication principle, ordered selections and factorials, choosing without regard to order and the binomial coefficient, and the number of paths in a tree with exactly k successes.
  13. Bernoulli's formulaLearningRepeated independent trials with success and failure, from a tree of three stages to Bernoulli's formula, when the formula fits and when it does not, and the probability of k successes in n trials.
  14. Problems with Bernoulli's formulaLearningAt least, at most and between two numbers, at least one success by the complement, the smallest number of trials, the most likely number of successes, Bernoulli's formula as a stage of a tree, and conditional probability with Bernoulli's formula.
  15. Practice: counting and Bernoulli's formulaPracticeExactly, at least and at most with Bernoulli's formula, conditional probability with Bernoulli's formula, ordered selections and the binomial coefficient, Bernoulli's formula as a stage of a tree, and the smallest number of trials.
  16. Review: probabilityPracticeA summary: the two-way table, the probability tree or Bernoulli's formula – by the experiment's structure; and questions on the whole chapter: the rules of probability and independence, a 3×2 table and the reverse question, a tree without putting back, Bernoulli's formula with conditional probability, and the smallest number of trials.
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