14C

Sample space graphs and tree diagrams

Listing outcomes of multi-stage experiments with a grid or a tree diagram, and using complements such as "at least one".

Grids for two stages

Rolling two dice gives $6 \times 6 = 36$ equally likely outcomes, best shown on a grid: one die across, the other down. Each cell can show the sum.

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Example 1●

Two dice are rolled. Find $P(\text{sum} = 7)$ and $P(\text{sum} = 8)$.

Sum $7$ is on the whole diagonal: $6$ cells, so $\dfrac{6}{36} = \dfrac16$.

Sum $8$: $5$ cells, so $\dfrac{5}{36}$.

Tree diagrams

For three or more stages, a tree lists every outcome: each stage branches into its possibilities. Tossing three coins gives $2 \times 2 \times 2 = 8$ outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Example 2●●

Three coins are tossed. Find the probability of exactly two heads, and of at least one head.

Exactly two heads: HHT, HTH, THH, so $\dfrac38$.

$P(\text{at least one head}) = 1 - P(\text{TTT}) = 1 - \dfrac18 = \dfrac78$

'At least one' is the complement of 'none'.

Flashcards

Outcomes for two dice

$36$

Outcomes for three coins

$8$

At least one

$1 - P(\text{none})$

Practice

Two dice are rolled. Find $P(\text{sum} = 7)$.
Find the probability of a double.
Find $P(\text{sum} = 8)$.
Find the probability of at least one six.
Three coins are tossed. Find $P(\text{exactly two heads})$.
Find $P(\text{at least one head})$ for three coins.
How many outcomes are there when two dice are rolled?
The letters A, B and C are arranged in a random order. Find the probability that the arrangement starts with A.
Two dice are rolled. Find $P(\text{the sum is odd})$.