Lesson 15.2 · 25 min

Tree Diagrams

Drawing a tree to keep track of two or more events, multiplying along the branches and adding down the ends.

In this lesson you will learn to…

Draw a tree diagram for two successive events
Label every branch with its probability
Multiply along branches to find the probability of a combined outcome
Add down the end column when more than one outcome counts as a success
Handle problems with and without replacement

The problem tree diagrams solve

Once you have two events, keeping track of every possibility in your head stops working. A tree diagram does the bookkeeping for you: every path through it is one possible way the world could turn out, and no path is ever missed.

Building one

Work left to right.

Draw the first split. One branch for each outcome of the first event.
Label each branch with its probability. Check they add to 1.
Draw the second split from the end of every first branch. Every first outcome must lead to a full set of second outcomes.
Label those too. This is the step where “with or without replacement” matters.
Write the combined outcome at the end of each path, and its probability, found by multiplying along the path.

Multiply along, add down

Two directions, two operations, and mixing them up is the classic mistake:

Along a path (left to right) means this happened and then that happened — so multiply.
Down the final column means this outcome or that outcome would do — so add.

“At least one” — the shortcut worth knowing

Questions that ask for the probability of at least one of something usually have several qualifying paths, and adding them all is slow and error-prone.

Instead, find the probability of none and subtract from 1:

$P(\text{at least one}) = 1 - P(\text{none})$

There is only ever one path with none of the thing, so this turns a three- or four-step calculation into a one-step one. With three or more events the saving is enormous.

With and without replacement

This is the detail examiners test hardest. Read the question for what happens to the first object:

Wording	Second set of branches
“…and replaces it”	Identical to the first set
“…and does not replace it”	Denominator drops by 1
“…takes two at once”	Same as without replacement
“…eats it”	Same as without replacement

When the denominator drops, the numerator only drops on the branches where you actually removed one of that colour — which is why the two second-level splits end up different from each other.

Key rules and formulas

Multiply along the branches
$P (A then B) = P (A) \times P (B)$
Travelling left to right along one path through the tree gives one combined outcome. Multiply the probabilities you pass.
Add down the outcome column
If several different paths would count as a success, work out each one and then add them together. This is where most marks are lost — read the question carefully to see how many paths qualify.
Each set of branches sums to 1
Every time the tree splits, those branch probabilities must add to 1. It is a free check: if a pair of branches reads $0.3$ and $0.6$ , something is wrong.
With or without replacement
If the object is put back, the second set of branches is identical to the first. If it is not put back, the second set has a denominator one smaller — and the numerator depends on which branch you came along.
The whole tree sums to 1
Add every outcome in the final column. It must come to exactly 1. Do this every time — it catches arithmetic slips instantly.

Worked examples

Worked example 1

Two spins, with replacement

A spinner has 3 red sections and 1 blue section. It is spun twice. Find the probability of getting exactly one red.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Without replacement

A bag holds 5 green counters and 3 white counters. Two are taken out, one after the other, without replacement. Find the probability that both are green.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

At least one

Using the same bag (5 green, 3 white, without replacement), find the probability of getting at least one white.

Try it yourself first, then reveal the solution one step at a time.

Practice: tree diagrams

9 questions

Nine questions, including a diagram you click on and a set of steps to put in order.

Question 1 of 9Click the diagram
This tree diagram is for tossing a coin twice. Click the branch that represents getting a tail on the second toss, after a head on the first.
Click the right part of the diagram. You can also tab to each part and press Enter.
Question 2 of 9Put in order
Put these steps for solving a tree-diagram question into the right order.
Put these in the right order. Drag them, or use the arrow buttons.
1. Draw the first split, one branch per outcome of the first event
2. Multiply along each path to get the probability of each combined outcome
3. Label those branches with their probabilities and check they add to 1
4. Draw the second split from the end of every first branch
5. Add the outcomes that count as a success
Question 3 of 9Type the answer
A spinner has 3 red sections and 1 blue section, and is spun twice. Find $P (red, then red)$ .
Your answer
Question 4 of 9Multiple choice
A bag has 5 green and 3 white counters. One is taken out and not replaced. If it was green, what is the probability the second counter is also green?
Question 5 of 9Complete the table
A bag has 5 green (G) and 3 white (W) counters. Two are taken without replacement. Complete the second-pick probabilities.
First counter P(green second) P(white second)
Green Row 1, P(green second) Row 1, P(white second)
White Row 2, P(green second) Row 2, P(white second)
Question 6 of 9Type the answer
Still with 5 green and 3 white, taken without replacement: find $P (both green)$ . Give your answer as a fraction in its simplest form.
Your answer
Question 7 of 9Step by stepChallenge
The "at least one" shortcut, step by step.
The probability that it rains on any given day is $0.3$ , and each day is independent of the last. Find the probability that it rains at least once over three days.
1. Which is quicker: adding every path with rain in it, or finding "no rain at all" first?
Question 8 of 9Choose all that apply
Select every statement that is true about tree diagrams.
Select every correct answer — there may be more than one.
Each split's branches must add up to 1Without replacement, the second branches are the same as the firstYou add the outcomes you want at the endAll the final outcome probabilities must add up to 1You multiply along a path
Question 9 of 9Type the answerChallenge
A box has 4 fruit sweets and 6 mint sweets. Two are taken without replacement. Find $P (one of each)$ , as a fraction in its simplest form.
Your answer

First counter	P(green second)	P(white second)
Green	Row 1, P(green second)	Row 1, P(white second)
White	Row 2, P(green second)	Row 2, P(white second)

In this lesson you will learn to…

The problem tree diagrams solve

Building one

Multiply along, add down

“At least one” — the shortcut worth knowing

With and without replacement

Key rules and formulas

Multiply along the branches

Add down the outcome column

Each set of branches sums to 1

With or without replacement

The whole tree sums to 1

Worked examples

Practice: tree diagrams