Tree Diagrams

Tree diagrams show events that happen one after another. Each path through the tree is one outcome, which makes it easy to see which probabilities to multiply and which to add.

Tree diagrams for successive events

Each set of branches from one point adds up to 1.
Multiply along a path for "and".
Add the results of separate paths for "or".

Example. A spinner lands on red with probability 0,3. It is spun twice.

$P (red, red) = 0, 3 \times 0, 3 = 0, 09$
$P (at least one red) = 1 - P (no red) = 1 - 0, 7 \times 0, 7 = 0, 51$

Two-stage situations. Thandi takes the bus with probability 0,6 and walks otherwise. If she takes the bus, she is late with probability 0,1; if she walks, 0,25.

$P (late) = 0, 6 \times 0, 1 + 0, 4 \times 0, 25 = 0, 16$

With and without replacement

With replacement: the probabilities are the same at every stage.

Without replacement: the numbers on later branches change.

Example. A bag has 3 red and 2 white counters. Two are drawn without replacement:

$P (one of each) = \frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{3}{4} = \frac{12}{20} = 0, 6$

Three draws. A bag has 4 black and 6 white counters. Three are drawn without replacement:

$P (all the same colour) = \frac{4}{10} \cdot \frac{3}{9} \cdot \frac{2}{8} + \frac{6}{10} \cdot \frac{5}{9} \cdot \frac{4}{8} = 0, 2$

For three stages, you only need to draw the branches you need.

Key terms

Branch: A line in a tree diagram showing one outcome and its probability.
Path: A route through the tree from start to end; one combined outcome.
Successive events: Events that happen one after another.

Common mistakes

Adding along a path instead of multiplying.
Branches from one point that do not add up to 1.
Keeping the same denominator without replacement.
Missing a path in 'one of each' questions.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

A spinner lands on red with probability 0,3. It is spun twice. Calculate:

(a) $P (red both times)$ (2)

(b) $P (at least one red)$ (2)

Worked solution

Hidden - try the question first

Solution

(a) $0, 3 \times 0, 3$
$= 0, 09$
(b) $1 - 0, 7 \times 0, 7$
$= 0, 51$

Question 2

5 marks total

routine

A bag contains 3 red and 2 white counters. Two counters are drawn without replacement. Calculate the probability that they are different colours.

Worked solution

Hidden - try the question first

Solution

Red then white: $0, 3 \times 0, 3$
White then red: $\frac{2}{5} \times \frac{3}{4} = \frac{6}{20}$
$\frac{12}{20}$
$= 0, 6$

Question 3

6 marks total

complex

A bag contains 4 black and 6 white counters. Three counters are drawn without replacement. Calculate the probability that all three are the same colour.

Worked solution

Hidden - try the question first

Solution

All black: $0, 3 \times 0, 3$
$= \frac{24}{720}$
All white: $\frac{6}{10} \times \frac{5}{9} \times \frac{4}{8}$
$= \frac{120}{720}$
$\frac{144}{720}$
$= 0, 2$

Question 4

9 marks total

problem-solving

Thandi takes the bus to school with probability 0,6 and walks otherwise. If she takes the bus, the probability that she is late is 0,1. If she walks, the probability that she is late is 0,25.

(a) Draw a tree diagram. (2)

(b) Calculate the probability that Thandi is late on a given day. (3)

(d) Calculate the probability that she is late on exactly one of two days. (2)

Worked solution

Hidden - try the question first

Solution

(a) First branches: bus 0,6, walk 0,4
Second branches: late 0,1 / on time 0,9 after bus; late 0,25 / on time 0,75 after walking
(b) $0, 6 \times 0, 1 = 0, 06$
$0, 4 \times 0, 25 = 0, 1$
$P (late) = 0, 16$
(c) $0, 16 \times 0, 16$
$= 0, 0256$
(d) $2 \times 0, 16 \times 0, 84$
$= 0, 2688$

FAQs

When do I multiply and when do I add in a tree diagram?

Multiply along a path ('and'). Add different paths ('or').

How do I check my tree diagram?

The branches from each point add up to 1, and all the final path probabilities add up to 1.

Do I have to draw every branch?

No. For long trees, draw only the paths you need, but make sure you have not missed any.

Tree diagrams for successive events

With and without replacement

Key terms

Common mistakes

Practice questions

FAQs

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