Independent events
- Events are independent if one does not affect the other, such as flipping a coin and rolling a dice.
- For independent events, P(A and B) = P(A) × P(B). P(head and a six) = ½ × 1/6 = 1/12.
Tree diagrams
- Each set of branches shows the outcomes of one event. The probabilities on each set of branches add up to 1.
- Multiply along the branches for 'and'. Add the results of different routes for 'or'.
- A bag has 3 red and 2 blue counters. One is taken and replaced, then another is taken. P(both red) = 3/5 × 3/5 = 9/25.
Without replacement
- If the first item is not put back, the probabilities on the second set of branches change: there is one fewer item.
- 3 red and 2 blue counters, taken without replacement: P(both red) = 3/5 × 2/4 = 6/20 = 3/10.
At least one
- P(at least one) = 1 − P(none).
- P(at least one six in two rolls) = 1 − 5/6 × 5/6 = 1 − 25/36 = 11/36.
Key terms
- Independent events
- Events where one does not affect the other.
- Tree diagram
- A diagram with branches showing the outcomes of events and their probabilities.
- Branch
- One line on a tree diagram, showing one outcome.
- With replacement
- The first item is put back before the next is chosen.
- Without replacement
- The first item is not put back, so the probabilities change.