Tree diagrams

A tree diagram shows two or more events, one after the other. Events are independent if one doesn't affect the other, such as two spins of a spinner.

1. Each set of branches from one point adds up to 1.
2. Multiply along the branches for "this and then that".
3. Add the results when more than one route gives what you want.

Example: P(red) = 0.3 on a spinner. It is spun twice. Find P(both red) and P(exactly one red).

• First spin: red 0.3, not red 0.7
  • After red: red 0.3, not red 0.7
  • After not red: red 0.3, not red 0.7

1. P(both red) = 0.3 × 0.3 = 0.09
2. Exactly one red: red then not red, or not red then red
3. 0.3 × 0.7 + 0.7 × 0.3 = 0.21 + 0.21 = 0.42

"At least one" means one or more. It's quickest to work out the opposite, "none", and subtract from 1.

P(at least one) = 1 − P(none)

Example: the spinner with P(red) = 0.3 is spun twice. Find P(at least one red).

1. P(no reds) = 0.7 × 0.7 = 0.49
2. P(at least one red) = 1 − 0.49 = 0.51

Check: P(both red) + P(exactly one red) = 0.09 + 0.42 = 0.51.

If an item isn't put back, the second probabilities change: there is one fewer item, and maybe one fewer of that colour. These are dependent events.

Example: a bag has 5 red and 3 blue counters. Two are taken without replacement. Find P(both red) and P(one of each colour).

• First: red 5/8, blue 3/8
  • After red: red 4/7, blue 3/7
  • After blue: red 5/7, blue 2/7

1. P(both red) = 5/8 × 4/7 = 20/56 = 5/14
2. Red then blue: 5/8 × 3/7 = 15/56
3. Blue then red: 3/8 × 5/7 = 15/56
4. P(one of each) = 30/56 = 15/28