Probability

Probability = favourable outcomes ÷ total outcomes, and the answer always lies between 0 and 1. Getting it right is almost entirely about counting the denominator honestly.

The standard sample spaces, worth memorising:

• One die → 6 outcomes.
• Two dice → 36 ordered outcomes, not 12 and not 21. (3, 5) and (5, 3) are different throws.
• One coin → 2; three coins → 8.
• A pack of cards → 52: four suits of 13; 26 red and 26 black; 12 face cards; 4 aces; 4 of every rank.

Three question types:

1. Direct count — dice, coins, a card, a ball from a bag. The denominator is the total number of items, never the number of unfavourable ones.
2. Without replacement — use combinations: two balls from a bag of n is nC2, and two red from r red is rC2. With replacement it would be (r/n)² instead, because the first ball goes back.
3. "At least one" — far easier as 1 − P(none).

The single most common mistake is a wrong denominator: dividing by the number of unfavourable balls, or treating two dice as 12 outcomes. Before you divide, say the total out loud. A second guard: if your probability comes out greater than 1, the denominator is definitely too small.

P = favourable/total · 0 ≤ P ≤ 1 · P(not A) = 1 − P(A) · one die 6, two dice 36, pack 52, three coins 8 · without replacement → nCr

A bag has 4 red and 5 white balls. Two are drawn at random without replacement. Find the probability that both are red. → Total ways = 9C2 = (9 × 8)/2 = 36. Favourable = 4C2 = (4 × 3)/2 = 6. Probability = 6/36 = 1/6. Sanity check by the step-by-step route: the first ball is red with probability 4/9, and then the second with 3/8, giving 4/9 × 3/8 = 12/72 = 1/6 — the same answer.