Probability & Counting

Probability covers Venn diagrams, tree diagrams, mutually exclusive and independent events, the addition and multiplication rules, and counting principles

Probability covers Venn diagrams, tree diagrams, mutually exclusive and independent events, the addition and multiplication rules, and counting principles including permutations and arrangements.

Key concepts

  • P(A or B) = P(A) + P(B) − P(A and B)
  • Independent: P(A and B) = P(A)·P(B)
  • Mutually exclusive: P(A and B) = 0
  • Counting: arrangements n!, with restrictions

How it works

Draw the diagram (Venn or tree) before calculating. Decide if events are independent or mutually exclusive. For counting, decide whether order matters and whether repetition is allowed. Most probability questions become routine once the information is organised. Fill in a Venn diagram from the inside out, starting with the overlap; complete every row and column total in a contingency table; and label every branch of a tree diagram with its probability before you multiply.

Step by step

1. Define events

Name events A, B, etc. and list known probabilities.

2. Diagram first

Venn or tree diagram to visualise overlaps.

3. Pick the rule

Addition for OR, multiplication for AND (independent).

4. Check

Probabilities must lie between 0 and 1.

Worked examples

P(A) = 0.4, P(B) = 0.5, P(A∩B) = 0.2. Find P(A∪B).

  1. P(A∪B) = 0.4 + 0.5 − 0.2 = 0.7

Answer: 0.7

How many 4-letter codes can be made from the letters A to F if no letter may repeat?

  1. 6 choices for the first letter, then 5, 4 and 3
  2. Multiply: 6 × 5 × 4 × 3

Answer: 360 codes

A bag has 3 red and 2 blue balls. Two are drawn without replacement. Find P(both red).

  1. P(first red) = 3/5
  2. P(second red, given the first was red) = 2/4
  3. Multiply along the tree: 3/5 × 2/4

Answer: 3/10

In a class of 40 learners, 25 play soccer, 18 play netball and 7 play both. How many play neither, and what is P(soccer only)?

  1. Soccer only = 25 − 7 = 18 and netball only = 18 − 7 = 11
  2. Neither = 40 − (18 + 7 + 11) = 4
  3. P(soccer only) = 18/40

Answer: 4 learners play neither; P(soccer only) = 9/20

P(A) = 0.3, P(B) = 0.6 and P(A and B) = 0.2. Are A and B independent?

  1. P(A) × P(B) = 0.3 × 0.6 = 0.18
  2. 0.18 ≠ 0.2

Answer: No, A and B are not independent

A coin is tossed three times. Find P(at least one head).

  1. P(no heads) = (½)³ = 1/8
  2. P(at least one head) = 1 − 1/8

Answer: 7/8

Common mistakes

  • Treating dependent events as independent
  • Counting an outcome twice in OR problems
  • Forgetting to subtract the intersection
  • Forgetting to subtract the overlap when filling in a Venn diagram.

Frequently asked questions

Are counting principles examined?

Yes — typically a final-section question on arrangements.

What is a contingency table?

A two-way table that shows how a group splits by two categories, such as grade and sport. It is used to find probabilities and to test whether two events are independent.

When do I subtract the overlap?

When events can happen together. P(A or B) = P(A) + P(B) − P(A and B) avoids counting the overlap twice; for mutually exclusive events the overlap is zero.

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