Tree Diagrams

Video coming soon3 worked examples

Theory

Probability trees are visual diagrams used to calculate the probabilities of different outcomes in sequential events (events that happen one after the other). They help visualise and compute probabilities in a clear, organised manner.

1. The Structure of a Tree Diagram

Each branch on the tree represents a possible outcome, with its associated probability written directly on the line.
The branches originating from a single, shared point represent mutually exclusive events, meaning only one of them can possibly occur at that time.
Because these branches cover all possible outcomes for that specific event, the probabilities on branches coming from the same point must always add up to exactly 1.

2. Calculating Probabilities (The Two Rules)

Once your tree diagram is drawn, you use two simple rules to find your answers:

The "AND" Rule (Multiply): To calculate the final 'end' probability of a specific sequence of events happening (e.g., getting a Head and then a Tail), you must multiply the probabilities along that specific path of branches.
The "OR" Rule (Add): If a question has more than one successful path (e.g., finding the probability of getting exactly one Head, which could be Head-Tail or Tail-Head), you calculate the final probability of each valid path and then add them together.

3. The "At Least One" Shortcut

In exams, you will frequently be asked to find the probability that an event happens "at least once" over multiple trials.

Calculating every single successful combination can be highly time-consuming. Instead, use the shortcut: find the probability that the event never happens at all (e.g., missing the train every single time), and subtract that answer from 1.

Worked examples

Example 1

Example 1: Independent Events

A commuter takes the bus to work. The probability that the bus is late on any given morning is 0.15. The commuter takes the bus on Monday and Tuesday.

(a) Draw a tree diagram to represent this information.

(b) Calculate the probability that the bus is late on both days.

(a) To draw this, start with two branches: 'Late' (0.15) and 'On Time' (0.85). From the end of each of those branches, draw another set of identical 'Late' (0.15) and 'On Time' (0.85) branches for Tuesday.

(b) To find Late and Late, we multiply along the branches:

$P(\text{Late, Late}) = 0.15 \times 0.15 = 0.0225$ .

$P(\text{Late, On Time}) = 0.15 \times 0.85 = 0.1275$
$P(\text{On Time, Late}) = 0.85 \times 0.15 = 0.1275$

Add the valid paths together: $0.1275 + 0.1275 = 0.255$ .

Example 2

Example 2: Dependent Events (Without Replacement)

A drawer contains 7 black socks and 5 white socks. A man reaches into the drawer in the dark and pulls out one sock. He keeps it, and then reaches in to pull out a second sock.

(a) Draw a tree diagram to show all possible outcomes.

(b) Calculate the probability that the man selects a matching pair of socks.

(a)

First Draw: The probability of Black is $\frac{7}{12}$ . The probability of White is $\frac{5}{12}$ .
Second Draw (Dependent): Because a sock has been removed, there are only 11 socks left.

If Black was drawn first, the second branches are Black ( $\frac{6}{11}$ ) and White ( $\frac{5}{11}$ ).
If White was drawn first, the second branches are Black ( $\frac{7}{11}$ ) and White ( $\frac{4}{11}$ ).

(b) A matching pair means (Black and Black) or (White and White).

$P(\text{Black, Black}) = \frac{7}{12} \times \frac{6}{11} = \frac{42}{132}$
$P(\text{White, White}) = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132}$

Add them together: $\frac{42}{132} + \frac{20}{132} = \frac{62}{132} = \frac{31}{66}$ (or approx 0.47).

Example 3

Example 3: The "At Least One" Shortcut

A quality assurance inspector at a factory tests 3 identical items chosen at random from a production line. The probability that any single item passes the inspection is 0.8.

Calculate the probability that at least one of the three items fails the inspection.

Instead of calculating all the combinations of 1, 2, or 3 failures, we can use the shortcut.

Find the probability that none of them fail (meaning they all pass).

$P(\text{Pass, Pass, Pass}) = 0.8 \times 0.8 \times 0.8 = 0.512$

Subtract this answer from 1.

$P(\text{At least one fail}) = 1 - 0.512 = 0.488$ .

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8 Tree Diagrams questions with answers and video solutions