Grade 11 Probability

Grade 11 probability: dependent and independent events, the product rule, Venn diagrams, tree diagrams and contingency tables, with worked CAPS examples.

Probability is about 20 of the 150 marks in Grade 11 Paper 1. Building on Grade 10 Venn diagrams, you now decide whether events are independent (one doesn't affect the other) or dependent, and you use the product rule P(A and B) = P(A) × P(B) for independent events.

You organise problems with Venn diagrams (including three events), tree diagrams for events that happen in stages, and two-way contingency tables, and you use them to test for independence. Grade 12 adds the fundamental counting principle.

Key concepts

  • Independent events: P(A and B) = P(A) × P(B)
  • Dependent events: the outcome of the first changes the second
  • Addition rule: P(A or B) = P(A) + P(B) − P(A and B)
  • Tree diagrams for events in stages, with and without replacement
  • Contingency tables to test whether two events are independent
  • Venn diagrams with two and three events

Worked examples

A and B are independent, with P(A) = 0.6 and P(B) = 0.3. Find P(A or B).

P(A and B) = 0.6 × 0.3 = 0.18. P(A or B) = 0.6 + 0.3 − 0.18 = 0.72.

A bag holds 4 red and 6 blue balls. Two are drawn without replacement. Find P(both red).

P = 4/10 × 3/9 = 12/90 = 2/15. After the first red ball is taken, only 3 of the remaining 9 balls are red.

In a survey of 200 learners, 80 are boys, 120 like soccer and 48 are boys who like soccer. Are 'boy' and 'likes soccer' independent?

P(boy) = 0.4 and P(soccer) = 0.6, so P(boy) × P(soccer) = 0.24. P(boy and soccer) = 48/200 = 0.24. They are equal, so the events are independent.

A fair coin is tossed three times. Find the probability of exactly two heads.

There are 8 equally likely outcomes. HHT, HTH and THH have exactly two heads, so P = 3/8.

Common mistakes

  • Confusing independent with mutually exclusive events.
  • Not reducing the totals on the second branch of a tree diagram for 'without replacement'.
  • Multiplying probabilities for events that are not independent.
  • Adding instead of multiplying along the branches of a tree diagram.

Exam tips

  • Multiply along the branches of a tree diagram; add the results for different paths.
  • To test independence, compare P(A and B) with P(A) × P(B) and write a conclusion.
  • Label every branch of a tree diagram with its probability.

Frequently asked questions

What is the difference between independent and mutually exclusive events?

Mutually exclusive events can't happen together, so P(A and B) = 0. Independent events can happen together; one just doesn't change the probability of the other, so P(A and B) = P(A) × P(B).

When do I use a tree diagram?

When events happen one after another, such as drawing two cards or tossing a coin several times.

What is a contingency table?

A two-way table that shows how a group splits by two characteristics, used to calculate probabilities and test for independence.

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