A tree diagram turns a wordy probability question into a picture where the arithmetic is obvious. The temptation is to shortcut it and multiply numbers straight from the question, which is where errors appear.
The method
- Draw the first set of branches and label each with its probability.
- Draw the second set from every first-stage branch, even the ones you think you will not need.
- Check each pair of branches sums to 1.
- Multiply along branches to find the probability of a full path.
- Add the relevant paths for the outcome asked about.
The 'at least one' shortcut
For 'at least one' questions, find the probability of none and subtract from 1. It is usually one calculation instead of three, and fewer calculations means fewer slips.
Conditional probability
When a question says 'given that', you are restricting attention to part of the tree. Add the paths that satisfy the condition, then divide the paths you want by that total. Writing the fraction out in words first prevents the usual mix-up over which total goes underneath.