Tree Diagrams for Probability

Why we need tree diagrams

When a problem has two or more events — flipping a coin twice, drawing two counters, taking two penalty kicks — listing every outcome in your head gets confusing fast. A tree diagram lays out every possible path clearly, with a probability on each branch, so you can work systematically and not miss any outcome.

The two golden rules

A tree diagram works on two simple rules:

1. MULTIPLY along the branches to find the probability of a complete path. 2. ADD the path probabilities of all outcomes that satisfy the question.

There is also a built-in check: the branches leaving any single point must add up to 1, because they cover every possibility at that step.

Worked example: flipping a coin twice

Each flip is head (H) or tail (T), both with probability 1/2. The tree has a first set of branches for flip 1, and from each of those, a second set for flip 2. That gives four end paths.

Path	Calculation	Probability
H then H	1/2 × 1/2	1/4
H then T	1/2 × 1/2	1/4
T then H	1/2 × 1/2	1/4
T then T	1/2 × 1/2	1/4

Check: the four paths add to 1/4 + 1/4 + 1/4 + 1/4 = 1. ✓

Now answer questions:

• P(two heads) = the H-H path = 1/4.
• P(exactly one head) = H-T path + T-H path = 1/4 + 1/4 = 1/2. (Here we add, because two different paths both give exactly one head.)
• P(at least one head) = 1 − P(no heads) = 1 − P(T-T) = 1 − 1/4 = 3/4.

That last trick — taking 1 minus the unwanted outcome — often saves work.

With replacement vs without replacement

This is where exam marks are won and lost. A bag holds 3 red and 2 blue counters (5 in total). We draw two counters.

With replacement: the first counter is put back before the second draw. The totals never change, so P(red) = 3/5 and P(blue) = 2/5 on both sets of branches. The events are independent.

• P(red then red) = 3/5 × 3/5 = 9/25.

Without replacement: the first counter is kept, so only 4 counters remain for the second draw, and the probabilities change depending on what was drawn first. The events are now dependent.

• After drawing a red, 2 reds and 2 blues remain (4 total), so the second-draw P(red) = 2/4.
• P(red then red) = 3/5 × 2/4 = 6/20 = 3/10.

Always ask whether the item is replaced — it completely changes the second set of branches.

Worked example: without replacement in full

Using the bag of 3 red and 2 blue, drawn without replacement, here are all four paths.

First	Second	Calculation	Probability
Red	Red	3/5 × 2/4	6/20
Red	Blue	3/5 × 2/4	6/20
Blue	Red	2/5 × 3/4	6/20
Blue	Blue	2/5 × 1/4	2/20

Check: 6/20 + 6/20 + 6/20 + 2/20 = 20/20 = 1. ✓

Now find P(one of each colour). Two paths give one red and one blue: Red-Blue and Blue-Red.

• P(one of each) = 6/20 + 6/20 = 12/20 = 3/5.

Activity: design your own tree

1. Put a few coloured items in a bag (for example 4 sweets: 2 green, 2 yellow).
2. Draw a tree for picking two without replacement, writing the changing fractions on each branch.
3. Multiply along each path and check all paths sum to 1.
4. Use the tree to find P(two the same colour) and P(one of each).
5. Then redo it with replacement and compare how the answers change.

Why this matters

Tree diagrams turn tangled multi-step probability into a clear, checkable picture, and they underpin everything from genetics and risk analysis to decision-making in games and finance. Make sure the single-event foundations are solid first with probability basics, and keep your fraction multiplication sharp with multiplying fractions.