What a tree is for
Some chance problems happen in stages: flip a coin, then roll a die; draw a card, then draw another. A probability tree is a diagram that lays the stages out left to right. Each stage splits into branches, one for each possible result, and each branch is labelled with its probability.
Trees do not replace the maths. They organise it, which is useful because most mistakes in multi-stage problems come from losing track of what has already happened.
The two rules
- Multiply along a path. The probability of a complete path from the start to a final outcome is the product of the branch probabilities on it.
- Add across paths. If several different paths lead to the same kind of result, add their probabilities.
Also, the branches leaving any single point must add up to 1, because something has to happen.
Example 1: coin then die
Flip a fair coin. If it lands heads, roll a fair die; if tails, stop. What is the chance of heads followed by a six?
- The first stage has two branches: heads (1/2) and tails (1/2).
- From heads, the second stage has six branches, each 1/6.
The path “heads, then six” has probability 1/2 × 1/6 = 1/12, about 8.3%.
Here the second stage does not depend on the first, since the die has the same chances whatever the coin did. This is the independent case, and it agrees with the multiplication principle.
Example 2: drawing marbles without putting them back
A bag has 3 red marbles and 2 blue marbles. You draw two marbles, one after the other, without replacement. What are the chances of each combination of colours?
First draw: red has probability 3/5; blue has probability 2/5.
Second draw, after a red first: the bag now holds 2 red and 2 blue, so red is 2/4 and blue is 2/4.
Second draw, after a blue first: the bag now holds 3 red and 1 blue, so red is 3/4 and blue is 1/4.
Now multiply along each path:
| Path | Calculation | Probability |
|---|---|---|
| Red, red | 3/5 × 2/4 | 6/20 = 3/10 |
| Red, blue | 3/5 × 2/4 | 6/20 = 3/10 |
| Blue, red | 2/5 × 3/4 | 6/20 = 3/10 |
| Blue, blue | 2/5 × 1/4 | 2/20 = 1/10 |
Check: 3/10 + 3/10 + 3/10 + 1/10 = 10/10 = 1. The totals must reach exactly 1; if they do not, an arithmetic slip has crept in.
To answer “one of each colour”, add the two paths that give one red and one blue: 3/10 + 3/10 = 6/10 = 3/5. The chance of two reds is 3/10 and of two blues is 1/10.
The branches on the second level change because the first draw changes the bag. That is the defining feature of dependent events, and the tree shows it directly: the probabilities on the second set of branches are conditional probabilities, one set for each first-draw outcome.
Example 3: a two-round game
Imagine a simple game where each round you win with probability 0.6 and lose with probability 0.4, and rounds are independent. This is purely an illustration of the method with made-up numbers. What is the chance of winning exactly one of two rounds?
- Win then lose: 0.6 × 0.4 = 0.24
- Lose then win: 0.4 × 0.6 = 0.24
Add the two paths: 0.24 + 0.24 = 0.48. The other paths are win-win at 0.6 × 0.6 = 0.36 and lose-lose at 0.4 × 0.4 = 0.16, and 0.48 + 0.36 + 0.16 = 1.00.
A recipe for building a tree
- List the stages in time order.
- For each stage, list the possible results.
- Label each branch with its probability, remembering to update the probabilities if earlier results change what is left.
- Multiply along each full path.
- Add the paths that match the question.
- Check that all final path probabilities total 1.
Where trees help most
- Problems where each stage changes the next, such as drawing cards or marbles.
- Questions that ask for a combination of results in a particular order.
- Problems where you want to see why a conditional chance differs from the original one, which is the bridge to Bayes theorem.
For many repeated identical stages, a tree becomes too large to draw. Then a formula, such as the binomial approach, does the same job more compactly, and a tree is best used to understand what the formula is counting. Trees also scale into decision trees, where some branches are choices and not chance events and the values at the ends are weighed together.
Pitfalls
- Not updating the second stage. The most common error is to keep 3/5 on the second draw even though a marble has gone.
- Adding along a path or multiplying across paths. The rules are the other way round.
- Dropping a branch. If the totals do not reach 1, a branch is missing.
- Treating a past result as changing an independent event. In the coin-and-die case, an earlier outcome does not change the later one.