Tree Diagrams

Flip a coin twice. Draw two counters from a bag, one after the other. Take a test, then take another. Whenever an experiment happens in stages, it's frighteningly easy to lose track of a possibility — "wait, did I count heads-then-tails and tails-then-heads?" A tree diagram fixes that. It maps every path the experiment could take, writes the probability on each branch, and never lets a single outcome slip through the cracks.

Think of it like a choose-your-own-adventure map: you start at one point, the world splits into the things that could happen first, then each of those splits into what could happen next. Follow any route from the start to a tip and you've traced one complete story of the experiment.

The method

A tree diagram draws each stage as a set of branches, with the probability written on every branch. Following one path from the start to an endpoint traces one complete outcome.

Two rules turn the picture into numbers:

Multiply ALONG the branches of a path — that gives the probability of this and then that. For the top path, 0.6 \times 0.7 = 0.42.
Add the path-probabilities for the different ways an event can happen — this way or that way.

And a built-in check: the branches at each split add up to 1 (0.6 + 0.4 = 1), and so do all four endpoint probabilities (0.42 + 0.18 + 0.08 + 0.32 = 1).

For a multi-stage experiment drawn as a tree:

the branch probabilities coming out of each split sum to 1 (they cover every possibility at that stage);
multiply ALONG a path (AND) to get the probability of that whole sequence of outcomes;
add BETWEEN paths (OR) to combine the separate ways an event can occur;
this works whether or not the stages are independent — if a later split depends on the earlier outcome, just read that split's branch probabilities off its own branches.

Worked example 1: two flips of a fair coin

Flip a fair coin twice. Stage 1 splits into H and T, each \tfrac{1}{2}. Stage 2 splits the same way again. There are four paths: HH, HT, TH, TT, each with probability \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}.

Exactly one head? That happens two ways — HT or TH — so add those paths:

P(\text{exactly one head}) = \tfrac{1}{4} + \tfrac{1}{4} = \tfrac{1}{2} = 0.5.

Multiply along each path, add between the paths you want. That's the whole game.

Worked example 2: without replacement (the branches change)

A bag has 3 orange and 2 lemon sweets. You eat one, then eat another — without replacement. The crucial thing: after the first sweet is gone, only 4 sweets remain, so the second set of branches uses new probabilities. If the first was orange, the second-pick chance of orange is \tfrac{2}{4}, not \tfrac{3}{5}.

Both orange? Multiply along that path: \tfrac{3}{5} \times \tfrac{2}{4} = \tfrac{6}{20} = 0.3.

Worked example 3: "at least one"

With the sweets above, what's the chance of at least one lemon? You could add up the three paths that contain a lemon — but there's a slicker trick. "At least one lemon" is the exact opposite of "no lemon at all" (i.e. both orange), and we just found P(\text{both orange}) = 0.3. So

P(\text{at least one lemon}) = 1 - P(\text{both orange}) = 1 - 0.3 = 0.7.

On a tree, "at least one" almost always means "1 minus the none path." It saves you adding three fiddly branches when subtracting one does the job.

Three traps snare almost everyone with tree diagrams — learn to spot all three:

Multiply along a path, add between paths. Along one route (this AND then that) you multiply. To combine several separate routes (this way OR that way) you add. Muddling these is the number-one tree mistake.
"Without replacement" changes the second branches. If you keep the first item, the second set of probabilities must use the updated counts (one fewer in the bag) — never the original ones.
Sanity-check every node. The branches leaving any single point must sum to 1. If they don't, you've dropped an outcome or mis-copied a fraction.

The little diagram you're drawing scales up into one of the most useful structures in the whole of applied maths. Doctors use probability trees to map how a diagnostic test's outcomes flow into "actually ill / actually healthy." Banks branch out possible market moves to price risk. And in machine learning, a "decision tree" is a core algorithm — it splits data again and again, exactly like your branches, to classify things (spam or not, cat or dog, fraud or fine).

The clever part? The rule that powers those grown-up models is the very same one you just learned: multiply along, add across. Chain thousands of these branches together and you can compute the odds of dizzyingly complex sequences of events. The humble two-flip coin tree and a real predictive AI are the same idea, just at different sizes.

The method

Worked example 1: two flips of a fair coin

Worked example 2: without replacement (the branches change)

Worked example 3: "at least one"

See it explained