Probability
What you must understand
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event) = number of favourable outcomes ÷ total outcomes. Know the complement rule P(not A) = 1 − P(A), the addition rule P(A or B) = P(A) + P(B) − P(A and B), and for mutually exclusive events P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
You should construct and use sample space diagrams, two-way tables, tree diagrams and Venn diagrams. Understand mutually exclusive (cannot happen together) versus independent (one does not affect the other's probability). Know experimental versus theoretical probability and the idea that relative frequency tends towards theoretical probability as trials increase.
Traps that appear in questions
- Adding probabilities for events that are not mutually exclusive without subtracting the overlap.
- Treating without replacement draws as independent; the second probability must be adjusted.
- Confusing "at least one" with "exactly one" - use 1 − P(none) for "at least one".
- Forgetting that probabilities of all outcomes sum to 1.
- Misreading tree diagrams: multiply along branches, add between branches.
Worked example
A bag has 4 red and 6 blue counters. Two are drawn without replacement. Find P(both red).
Answer: First draw P(red) = 4/10. After removing one red, 3 red remain out of 9, so P(second red) = 3/9. Multiply: 4/10 × 3/9 = 12/90 = 2/15 ≈ 0.133.
Rule of thumb: "and" usually means multiply; "or" usually means add - but check for overlap and replacement.