Probability Trees
A probability tree is a diagram that shows all the possible outcomes of two or more events as branches, with the probability of each individual outcome written on its branch. You multiply along the branches to find the probability of a sequence of outcomes, and add together the probabilities of any separate routes that both lead to the outcome you want.
Method
- Draw each stage of the tree as a set of branches leading out from a point, making sure the probabilities on any set of branches from the same point add up to 1.
- Decide whether the events are independent (the second set of branches stays the same) or dependent, such as without replacement (the second set of branches changes based on the first outcome).
- To find the probability of one specific sequence of outcomes, multiply the probabilities along that single path through the tree.
- To find the probability of two or more different outcomes that would satisfy the question, work out the probability of each matching route separately, then add them together.
- For at least one questions, it is often quicker to find the probability that the event does not happen at all, then subtract this from 1.
- Check your final answer is sensible: all the probabilities for every complete route through the tree should add up to 1.
Worked example
A biased spinner has a probability of 0.3 of landing on red on any spin. The spinner is spun twice. Work out the probability that the spinner lands on red exactly once.
- P(not red) = 1 - 0.3 = 0.7.
- The two routes that give exactly one red are: red then not red, or not red then red.
- P(red then not red) = 0.3 x 0.7 = 0.21, and P(not red then red) = 0.7 x 0.3 = 0.21.
- Add the two routes: 0.21 + 0.21 = 0.42.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
A biased coin has P(Heads) = 0.65. The coin is flipped twice. Calculate the probability of getting exactly one Head.
A box contains 7 milk chocolates and 5 dark chocolates only. Two chocolates are taken at random, one after another, without replacement. (a) Work out the probability that both chocolates are milk chocolates. (2) (b) Work out the probability that at least one of the chocolates is dark. (2)
A school has two photocopiers, Copier 1 and Copier 2, which break down independently of each other on any given day. On a randomly chosen day, the probability that Copier 1 breaks down is 0.12, and the probability that Copier 2 breaks down is 0.06. (a) Find the probability that both copiers break down on the same day. (2) (b) Find the probability that exactly one of the two copiers breaks down on that day. (3)
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