Probability Trees: Mapping Every Possible Outcome
Two drawing rules, one tidy map of "what could happen"
By Daon Opus · Updated September 22, 2026
Last Saturday we planned a picnic. The forecast said a 30% chance of rain and the grill had a habit of refusing to start — two events, four possible afternoons. Instead of guessing, my daughter drew a little tree: dry or wet, then grill works or grill breaks, with every path traced to a leaf. Four leaves, every possibility on one page, nobody arguing about what could happen. That is a probability tree — the family craft of turning a tangled "what if?" into a map you can read at a glance. This guide draws that map branch by branch. It stays hands-on: the And/Or formulas and the counting shortcuts that skip the drawing are follow-ups of their own, and the deeper question of which numbers to trust lives in the experimental vs. theoretical probability guide. Here, we draw.
What a Probability Tree Is
A probability tree starts at a single point — the moment before anything happens. From that trunk, branches grow: one branch per possible result of the first event. Each branch gets its probability written on it. Then from the tip of every branch, the next event forks again. A two-event tree with two results each grows four leaf-branches, and the leaves at the far right are the full list of what can happen.
Rain? (30% / 70%) → Grill? (80% / 20%) → four leaves:
dry+grill, dry+break, wet+grill, wet+break
The whole point of the tree is completeness. People guess at "maybe it rains, maybe it doesn't, and the grill — ugh" and forget a branch. A tree has a rule built in at every fork that prevents missing outcomes, and that rule is the first one to learn.
Drawing Rule 1: Branches at a Fork Add to 1
At any fork, the probabilities on the branches must add up to 1 (or 100%). Something has to happen, so the "is it raining?" fork has a dry branch and a wet branch, and their chances must cover every possibility between them. If you have written a 60% branch, the missing one is forced: 40%. This rule quietly catches mistakes — a fork whose branches total 1.3 is a panic in the making, not a prediction.
When the outcomes at a fork are equally likely — a coin, a fair die, a shuffled deck — just count them. Two sides means 1/2 each, six faces means 1/6 each. Unequal chances, like weather, come from experience or a forecast. Either way the fork still sums to 1, and the tree does not care where the numbers came from.
Drawing Rule 2: Multiply Along a Path
Once every branch is labeled, reading the tree is one operation:to get the probability of a whole path, multiply the probabilities written along its branches. Follow rain → grill-breaks: 30% × 20% = 6%. Follow dry → grill-breaks: 70% × 20% = 14%. Those four numbers — 6, 14, 24, 56 — add up to 100%, just like the whole afternoon adds up to one Saturday.
Flip two coins: 1/2 × 1/2 = 1/4 for each of four leaves.
Or: one 40% chance and one 30% chance = 0.4 × 0.3 = 0.12 = 12%.
Why multiply? Because a path asks for two things at once — "rain and the grill breaks" — and when two independent things both need to happen, their chances pile up by shrinking the space each time. The tree turns that abstract pile-up into a literal walk along a path. No memorized formula; just follow the branches.
When the Second Fork Changes: With or Without Replacement
The tree earns its keep the moment the second event is not independent of the first. Picture a bag with 3 red and 2 blue marbles, picking twice:
- With replacement: you return the first marble, so the second draw always starts from 3 red + 2 blue — every fork at the second stage repeats the same 3/5 and 2/5.
- Without replacement: the bag changes. If the first draw was red, the second stage is 2 red + 2 blue (2/4 and 2/4). If the first was blue, it is 3 red + 1 blue (3/4 and 1/4).
The tree makes this visible in a way a formula never will: the second-stage branches are different numbers depending on which first branch you came down. That changing label is the entire concept of "the bag remembers" — and forgetting to update it is the classic marble-tree error. Always ask "after the first pick, how many of each are left?" before labeling the second fork.
Reading Backward: The "At Least One" Trick
Some questions are easier answered by what does not happen. "What's the chance of at least one head in three coin flips?" the tree answers by its opposite: the only path with no heads is tails-tails-tails. Rather than adding every path that contains a head, compute 1/8 and subtract from 1: 1 − P(none). Same for the picnic — "chance the grill works at least once this summer" is 1 minus the chance it fails every time. Because the leaves already cover all of "anything can happen," removing the one path you do not want leaves exactly what you do. It is subtraction instead of a long addition, and on a tree the "none" path is easy to see.
Slip Points That Predict Wrong Answers
- Multiplying what should be added. Two branches at the same fork that both lead to a win get added (dry grill-OK + wet grill-OK). Multiplication is for one path, addition is for collecting paths.
- Stale second-stage labels. Without replacement, the second fork's numbers must be re-counted after the first card/marble is removed. A tree with identical second forks has secretly assumed replacement.
- Forgetting to label branches. A tree with probabilities only at the leaves hides where the numbers came from. Label every branch, then the leaves become self-checking.
My first marble tree looked beautiful and answered every path wrong, because I drew the second fork as 3/5 and 2/5 twice — I was writing "with replacement" without realizing the problem meant without. The fix is now my standard question: "did the bag change?" Ask it at every fork and the tree does the rest.
Quick Practice Problems
- Flip two coins. What is the probability of exactly one head?
Answer: 1/2 — the HT and TH leaves: 1/4 + 1/4.
- 40% chance of rain, and if it rains, 50% chance the game is cancelled. Chance of a cancelled game?
Answer: 0.4 × 0.5 = 0.2 = 20%.
- A bag has 4 red and 1 blue marble. Draw twice without replacement. Chance that both are red?
Answer: (4/5) × (3/4) = 0.6 = 60% — the second fork refills to 3 red of 4.
- In three coin flips, chance of at least one head?
Answer: 1 − (1/8) = 7/8.
- Flipping one coin: do the fork labels add to 1?
Answer: Yes — 1/2 + 1/2 = 1, as every fork must.
Frequently Asked Questions
Why draw a tree when a formula is faster?
The tree is the thinking tool. Formulas tell you to multiply or "skip counting," but a tree shows which branches exist, which birthdays are missed, and which labels you forgot. Once the picture is right, the arithmetic is mostly multiplication.
Can a tree have more than two stages?
Yes — flip three coins and the tree has three forks, eight leaves. The rules never change: every fork sums to 1, every path multiplies. Trees grow comb-shaped the deeper they go, which is exactly why the counting shortcut (statistics and probability explained) earns its keep later.
What if the branches at a fork are not equally likely?
Fine — the tree does not need equal chances. A 30% dry branch followed by a 20% grill-break branch multiplies to 6% just like coin faces do. Equal-likelihood is a nice case, not a requirement.