Probability: Classical Definition, Dice, Coins and Cards
Probability is the short, high-scoring chapter: one or two 1–2 mark questions on coins, dice, cards or a bag of balls.
Definition and rules
For equally likely outcomes, P(E) = number of favourable outcomes / total number of outcomes. Then 0 ≤ P(E) ≤ 1; an impossible event has probability 0, a sure event 1, and P(E) + P(not E) = 1, so P(not E) = 1 − P(E). One die has 6 outcomes, two dice 36 ordered outcomes, and n coins give 2ⁿ outcomes.
Standard deck: 52 cards, 26 red and 26 black, four suits of 13, twelve face cards (J, Q, K in each suit) and four aces.
Worked example
Two dice are thrown together. Find the probability that the sum is 7, and that the sum is not 7. Favourable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 out of 36. P(sum 7) = 6/36 = 1/6; P(not 7) = 1 − 1/6 = 5/6.
Setting out
List the sample space or the favourable cases explicitly, count, then simplify the fraction. For "at least one" questions, use the complement rather than counting directly.
Traps that cost marks
- Counting "at least one" directly instead of using 1 − P(none) wastes time and causes errors.
- Aces are not face cards; a pack has 12 face cards, not 16.
- For two dice, (1, 6) and (6, 1) are different outcomes.
- Probabilities of mutually exclusive events add, but the total cannot exceed 1; never add probabilities of overlapping events.
- Leave the answer as a simplified fraction unless a decimal is demanded.