Sample spaces and combined events

Sample space grids

When two things happen together, list every combination in a grid.

Each cell is one equally likely outcome, so count cells, not different totals. Common slips: the 11 totals of two dice are not equally likely (12 happens 1 way out of 36); with two coins, HT and TH are different outcomes; a dice and a coin together give 6 × 2 = 12 outcomes, not 6 + 2.

Events are mutually exclusive if they cannot happen at the same time (a dice cannot show 2 and 5 at once). For mutually exclusive events, P(A or B) = P(A) + P(B): with P(A) = 0.25 and P(B) = 0.35, P(A or B) = 0.6. Multiplying is for A and B with independent events, not for "or".

When one of the two things is biased, use its own probabilities and multiply by the other's for each cell: a spinner with P(3) = 0.5 and a fair dice showing less than 3 (2/6) give 0.5 × 2/6 = 1/6 for that combination. Add the combinations that fit.

1. spinner A: 1, 3, 5; spinner B: 2, 4
2. outcomes: 3 × 2 = 6
3. totals: 3, 5, 5, 7, 7, 9
4. P(total greater than 5) = 3/6 = 1/2

A game is fair when each player has the same chance of winning. With two spinners numbered 1 to 4, an odd product needs both numbers odd: 2 × 2 = 4 of the 16 outcomes. The player who wins on odd has probability 4/16 = 1/4, so the game is not fair.