Basic probability

Favourable outcomes over total outcomes: When outcomes are equally likely, the probability of event A is the number of favourable outcomes ÷ the total number of outcomes. Every probability lies between 0 and 1.

Probabilities never exceed 1: If you get a probability above 1 or below 0, you've made an error — recheck your counts.

P(not A) = 1 − P(A): The complement of A is 'A does not happen', with P(A′) = 1 − P(A). This is the fast route to 'at least one' questions: P(at least one) = 1 − P(none).

'At least one' → do the opposite: Counting every 'at least one' case is slow; 1 − P(none) is almost always faster.

List every equally likely outcome in a grid: A sample space diagram (grid or list) shows every equally likely outcome. For two dice there are 36 outcomes; count the ones matching the event and divide by 36.

Order matters in the grid: (2,5) and (5,2) are different cells — count ordered outcomes so the total is 36.

Multiply along the chain — and update the totals: For events one after another, multiply the probabilities along the way. Without replacement, the counts change after each draw (one fewer item, and one fewer of that type).

Reduce the totals each time: After the first draw there are 9 pens left, not 10 — forgetting to reduce both totals is the classic slip.