Probability: including using tree diagrams

The idea

A tree diagram is a branching possibility space for combined experiments. Each branch shows one outcome at a stage, and the probability on that branch is written beside it. For independent stages, multiply along a path to get the probability of that sequence; for example, two fair coin tosses give P(heads then heads) = 1/2 × 1/2 = 1/4. For dependent stages, adjust the second-stage probabilities to reflect what has already been removed. Add the probabilities of all paths that satisfy the event. Tree diagrams also help you see whether outcomes are equally likely and whether all possibilities are covered. You may be asked to complete missing branches or probabilities, or to read a value from a finished diagram.

What earns the marks

• A tree diagram represents a combined experiment as successive stages, with each branch showing one outcome at that stage.
• The probability of a path is found by multiplying the probabilities along its branches.
• For independent stages, the probability on a second-stage branch is unchanged by the first-stage outcome.
• For dependent stages, second-stage probabilities must be recalculated after the first stage, for example when items are removed without replacement.
• The probability of an event made up of several paths is the sum of the probabilities of those paths.
• A completed tree diagram should have branch probabilities that sum to 1 at each set of branches from a node.

Common mistakes and how to put them right

• Adding along a path instead of multiplying.

  Put it right: The correct method is to multiply the probabilities on the branches of one path to obtain the probability of that sequence.

• Using the same second-stage probabilities when items are removed without replacement.

  Put it right: The correct method is to reduce the denominator and, where relevant, the numerator to match what remains.

• Forgetting to add all relevant paths for an event.

  Put it right: The correct method is to identify every path that satisfies the event and sum their probabilities.

• Assuming branches must be equally likely.

  Put it right: The correct method is to read or calculate each branch probability from the information given.

Exam tips

• Label each branch with its outcome and write the probability clearly beside the branch before calculating.
• For without-replacement questions, write the reduced fractions on the second set of branches so the change is visible.
• Check that the probabilities on branches from any single node sum to 1; this catches many slips.
• When an event has more than one path, list the paths first and then add their probabilities.