Mathematics of Chance: Equally Likely Outcomes
What the exam tests
Chance questions ask for probabilities from dice, coins, cards, bags and simple geometric regions. The key skill is listing or counting the sample space accurately and identifying favourable outcomes without double counting.
Core rules
- For equally likely outcomes, P(E)=number of favourable outcomes/total number of outcomes.
- 0 ≤ P(E) ≤ 1; P(not E)=1−P(E).
- Two dice have 36 ordered outcomes; one die has 6.
- A standard pack has 52 cards: 4 suits of 13, with 12 face cards (J, Q, K in each suit).
- With replacement, totals stay the same; without replacement, both favourable and total counts change.
- For a point chosen in a region, probability = favourable area/total area.
Worked example
Two dice are thrown. Find the probability that the sum is 9. The favourable ordered pairs are (3,6), (4,5), (5,4) and (6,3), so there are 4 favourable outcomes out of 36. P(sum 9)=4/36=1/9. The pair (6,3) is different from (3,6) because the dice are distinct.
Traps
Do not use 12 as the total for two dice. Treat (3,6) and (6,3) as separate outcomes. For “at least one” questions, the complement is often shorter. Face cards exclude aces. In without-replacement problems, reduce the denominator after each draw. Geometric probability compares areas or lengths of the same kind, not a length with an area.