Probability
Counting principles, tree diagrams, Venn diagrams
Counting Principles and Probability
The fundamental counting principle: if event A can occur in m ways and event B in n ways, then A and B together can occur in m × n ways. We also use tree diagrams, Venn diagrams and contingency tables to organise probability problems.
Example
Fundamental Counting Principle
A restaurant offers 3 starters, 5 mains and 4 desserts.
Total meal combinations = 3 × 5 × 4 = 60
A 4-digit PIN using digits 0–9 (repetition allowed):
Possibilities = 10 × 10 × 10 × 10 = 10 000
Example
Venn Diagram
In a class of 30: 18 like soccer (S), 12 like netball (N), 5 like both.
Only S = 18 − 5 = 13
Only N = 12 − 5 = 7
Neither = 30 − 13 − 5 − 7 = 5
P(S or N) = 25/30 = 5/6
Note
Remember
P(A or B) = P(A) + P(B) − P(A and B). If events are mutually exclusive, P(A and B) = 0. If independent, P(A and B) = P(A) × P(B). Tree diagrams: multiply along branches, add between branches.
Key Vocabulary
Counting principleMultiply the number of choices for each step
Venn diagramOverlapping circles showing relationships between sets
Mutually exclusiveEvents that cannot happen at the same time
Independent eventsEvents where one does not affect the other
Complementary eventsEvents that together cover all possibilities: P(A') = 1 − P(A)
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Counting principle
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