Why draw a tree
When an outcome depends on several steps, such as drawing cards one after another or a sequence of random events in a game, it is easy to lose track of what is being multiplied by what. A probability tree is a diagram that keeps each step visible. Each fork is a point where something random happens, each branch is one possible result, and each branch carries the chance of that result given everything that came before.
Trees are the visual form of conditional probability. Each number on a branch after the first step is a conditional chance, and reading across a path gives the chance of that whole sequence.
The two rules
Everything about a tree comes down to two rules.
- Multiply along a path. The chance of a full sequence is the product of the branch chances along it. This is the multiplication rule for dependent events: P(A then B) = P(A) × P(B | A).
- Add across paths. If several different paths all lead to the outcome you care about, add their chances. Paths are mutually exclusive, so no path is counted twice.
A useful check: the branches leaving any single fork must add to 1, and the chances of all the complete paths must also add to 1. If they do not, a number is wrong.
A worked example: two draws from a bag
A bag holds 3 red marbles and 2 blue marbles. You draw two without putting the first back. The numbers here are simple and chosen so the arithmetic is easy to verify.
First fork. Red with chance 3/5, blue with chance 2/5.
Second fork, after a red. 2 red and 2 blue remain among 4, so red is 2/4 and blue is 2/4.
Second fork, after a blue. 3 red and 1 blue remain among 4, so red is 3/4 and blue is 1/4.
Now multiply along each of the four paths:
| Path | Calculation | Chance |
|---|---|---|
| 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 |
Add the four chances: 3/10 + 3/10 + 3/10 + 1/10 = 10/10 = 1. The tree is consistent.
Using the tree to answer questions
Chance of exactly one red. Two paths give this: red-then-blue and blue-then-red. Add them: 3/10 + 3/10 = 6/10 = 3/5.
Chance the second marble is red. Two paths end in red: red-red and blue-red. Add: 3/10 + 3/10 = 3/5. Interestingly, that matches the chance that the first marble is red, 3/5. With no information about the first draw, each position is equally likely to be red. The tree makes this easier to see than it is by intuition. This way of combining the cases is exactly the law of total probability.
Chance the first was red, given the second was red. This reverses the order, which a tree makes manageable. Restrict attention to the paths where the second is red: red-red (3/10) and blue-red (3/10). The total is 6/10. Of that, the red-red path accounts for 3/10, so the conditional chance is (3/10) / (6/10) = 1/2.
That last step is Bayes theorem in practice: pick the paths that match what you observed, then compute the share belonging to the hypothesis of interest.
Trees for independent steps
Trees work for independent events too; the branch chances just stay the same at every fork. Flip a fair coin twice and each fork is 1/2 for heads and 1/2 for tails, giving four paths of 1/4 each. For longer sequences, though, trees get large quickly: three coin tosses give 8 paths, five give 32. For long runs of identical independent steps it is usually better to count outcomes using a formula, such as in the binomial distribution, and use a tree only for the first two or three steps to understand the structure.
You can prune a tree too. If you only care about whether something happens at least once, group all the other branches together into a single “none of these” branch. The complement rule often shortens the work: the chance of at least one success is 1 minus the chance of none.
Common mistakes
- Forgetting to update the second fork. In draws without replacement, the branch chances after the first step must reflect what has been removed. Using the original chances again treats the draws as independent when they are not.
- Adding along a path or multiplying across paths. Remember: multiply along, add across.
- Branches that do not sum to 1. Every fork must cover all the possibilities. If there is a third option, such as a draw of a different colour, it needs its own branch.
- Mixing up which condition applies. The number on a second-step branch is conditional on the path to that point, not on the other branches at the same level.
Trees and gaming
Trees are helpful for working through the structure of small random mechanics, such as a two-stage loot roll or a sequence of card draws, because they show each stage’s contribution. They explain what the numbers are; they do not change them. In a casino setting, every bet still carries a house edge, results of independent plays do not influence each other, and no diagram makes outcomes certain. Gambling carries real risk of harm, so if it feels hard to control, contact a local support service.
A good habit is to sketch a tree before reaching for a formula. If the tree stays small, you may not need the formula at all, and if it grows large, the diagram will still tell you which formula you need.