Probability Tree Generator

A probability tree is the standard way of teaching combined events, and the two things students get wrong are always the same: multiplying along a path rather than adding, and remembering that all the leaves must add up to one. This expands any number of stages into every path, shows the arithmetic, and reports the total — which had better be 100%, and is checked before you see it.

Stages and branches

One stage per line, as “Stage name: Branch 0.6, Branch 0.4”. Probabilities that do not add to 1 are rescaled.

Every path through the tree and its probability
Path	Probability
Positive → Positive	76.50%
Positive → Negative	13.50%
Negative → Positive	8.50%
Negative → Negative	1.50%

4 outcomes, adding to 100.0%.

Probabilities are multiplied along each path, which assumes the stages are independent of each other. Where a later stage depends on an earlier one — a second test whose accuracy changes given the first — those branch probabilities have to be the conditional ones, and you have to supply them.

What this generator does

Does the multiplication and the bookkeeping. The probability question generator here sets practice questions; this works out a specific tree you have described. The decision tree generator next door enumerates structure without probabilities — this one attaches a chance to every branch and produces the full outcome table, with the total shown so an error is immediately visible.

How to use this tool

Write one stage per line, as “Stage name: Branch 0.6, Branch 0.4”.
Add as many stages as you need.
Read the outcome table — each path's probability is the product along it.

Understanding the controls

Stages and branches: One stage per line: a name, a colon, then the branches with their probabilities. Branches that do not add to 1 are rescaled and the rescaling is stated, rather than silently producing a tree that sums to 90%.

Worked examples

Coin then weather: Four outcomes; heads-and-rain is 0.5 × 0.3 = 0.15.
Three stages of three branches: 27 outcomes, summing to 100%.
Branches entered as 0.3 and 0.3: Rescaled to 0.5 and 0.5, with a note saying so.
A 90% test applied twice: Both positive is 81%, not 90% — the point of the tool.

Common use cases

Teaching combined and conditional probability with a worked example
Checking a two-stage or three-stage probability by hand
Producing a table of outcomes for a test or worksheet
Working out the chance of a specific sequence of events
Showing why a 90% test applied twice is not 90%

How this generator works

Each stage's branches are combined with every path so far, and each path's probability is the product of the branch probabilities along it. That multiplication is only correct when the stages are independent of one another, which is stated on the page — where a later stage depends on an earlier one, the branch probabilities have to be the conditional ones and you have to supply them. The finished tree is checked before display: every path multiplies out to its own stated probability, the number of leaves matches the branching, and the leaves sum to exactly one.

Randomness and fairness

Nothing here is random. It is exact arithmetic over the tree you describe, and the same stages always produce the same table.

For how randomness is produced across the whole site, see how Generate Random works.

Assumptions this tool makes

The branches of each stage are exhaustive and mutually exclusive, and each stage's probabilities are as given.

Limitations and good to know

It multiplies along paths, which assumes each stage is independent of the ones before it. For genuinely conditional stages you must enter the conditional probabilities yourself.
It does not draw a branching diagram; the output is the complete outcome table.
It does not compute reverse conditional probabilities, so Bayes' theorem questions need working from the table by hand.
The tree is not stored between visits — copy the outcome table into the worksheet or notes it is for.

Common mistakes

Adding along a path instead of multiplying: Along a path you multiply; across the leaves you add. The table shows both so the difference is visible.
Using unconditional probabilities for a dependent second stage: If the second stage's chances change depending on the first, these are the wrong numbers. Enter the conditional ones.
Not checking that the leaves add to one: The total is displayed for exactly this reason. Anything other than 100% means a branch is wrong.

Practical tips

Build the two-stage version first and check it by hand before adding a third.
Use it to show why repeated 90%-accurate tests are not 90% accurate together — it is the most persuasive example there is.

Privacy and your data

The stages you type stay in your browser. Nothing is uploaded, nothing is stored between visits, and none of it is written into the page address. Analytics records only that the tool was used.

Frequently asked questions

Why do the branch probabilities get rescaled?: Because the branches of a stage must cover every possibility and therefore add to one. If yours add to 0.6, something is missing — the tool rescales so the arithmetic works and tells you it did, so you can go and find it.
Does this handle conditional probability?: It multiplies along paths, which is correct for conditional trees provided you enter the conditional probabilities for later stages. It cannot work them out for you, and the page says so.
How is this different from the decision tree generator?: That one enumerates structure — every path, no numbers. This one attaches probabilities and works out how likely each outcome is.