Elementary Probability

Lessons

Senior 1 · Mathematics

Elementary Probability

5 lessons

Lesson 1 of 5

Experiments, outcomes and sample spaces

22 min

Heads or tails at Amahoro Stadium

Before a football match at Amahoro Stadium, the referee tosses a coin and the captains call "heads" or "tails". Nobody can say which side will land up. That is exactly why the toss is fair. But everybody knows what can happen: heads or tails, nothing else. Probability starts here, by listing every possible result of an action whose result is uncertain. This lesson gives you the words for that list.

Definition

Experiment (random experiment): an action whose result cannot be predicted for certain, although all its possible results can be listed. Each time the experiment is done is called a trial.

Outcome: one possible result of an experiment.

Example: rolling a die is an experiment; "the die shows 4" is one of its outcomes.

Definition

Sample space: the set of all possible outcomes of an experiment.

Notation: the sample space is usually called $S$ , and $n (S)$ is the number of outcomes in it, as with the sets of Unit 1.

Example: for tossing a coin, $S = \{H, T\}$ and $n (S) = 2$ .

Each experiment has a list of possible results, its sample space.

Definition

Event: a set of outcomes, that is, a subset of the sample space. An event is named with a capital letter, and it happens (occurs) when the outcome of the trial is one of its elements.

Example: when you roll a die, the event "an even number" is $E = \{2, 4, 6\}$ , with $n (E) = 3$ . If the die shows 4, then $E$ has happened.

Worked example 1: listing sample spaces

Write the sample space and $n (S)$ for each experiment: (a) a card is picked from five cards numbered 1 to 5; (b) a learner is chosen from Uwase, Mugisha and Keza to lead a group; (c) a day of the week is chosen for a class trip.

Solution

$S = \{1, 2, 3, 4, 5\}$ , $n (S) = 5$ .
$S = \{\text{Uwase},\ \text{Mugisha},\ \text{Keza}\}$ , $n (S) = 3$ .
$S = \{\text{Mon},\ \text{Tue},\ \text{Wed},\ \text{Thu},\ \text{Fri},\ \text{Sat},\ \text{Sun}\}$ , $n (S) = 7$ .

Worked example 2: events as subsets

A die is rolled. List these events and count their outcomes: $A$ : "a number greater than 4"; $B$ : "a prime number"; $C$ : "a number less than 7"; $D$ : "the number 8".

Solution

$A = \{5, 6\}$ , $n (A) = 2$ .
$B = \{2, 3, 5\}$ , $n (B) = 3$ (1 is not a prime number).
$C = \{1, 2, 3, 4, 5, 6\} = S$ , $n (C) = 6$ : every outcome is in $C$ .
$D = \varnothing$ , $n (D) = 0$ : no outcome is in $D$ .

Two things at once

Some experiments have two stages, such as tossing two coins or rolling two dice. A tree diagram lists the outcomes stage by stage: each branch of the first stage splits into every result of the second.

Two coins: $2 \times 2 = 4$ outcomes. HT (the 100 Frw coin shows heads) is a different outcome from TH.

Important

Counting outcomes of a two-stage experiment. If the first stage has $m$ outcomes and the second has $n$ , the whole experiment has

m \times n

outcomes. Why: each of the $m$ first results is paired with each of the $n$ second results, like the Cartesian product $A \times B$ of Unit 1, which has $n(A) \times n(B)$ ordered pairs. So two coins give $2 \times 2 = 4$ outcomes, a coin and a die give $2 \times 6 = 12$ , and two dice give $6 \times 6 = 36$ .

For two dice, a tree would need 36 branches, so use an outcome grid instead. Each outcome is an ordered pair (first die, second die), and each cell of the grid below shows the total of the pair.

Rolling two dice: 36 outcomes. Each cell shows the total; the six ways to make 7 are shaded.

Worked example 3: events with two coins

Two coins are tossed. List the events (a) $E$ : "exactly one head", (b) $F$ : "at least one head", (c) $G$ : "no heads".

Solution From the tree, $S = \{HH, HT, TH, TT\}$ .

$E = \{HT, TH\}$ , $n (E) = 2$ .
"At least one" means one or more: $F = \{HH, HT, TH\}$ , $n (F) = 3$ .
$G = \{TT\}$ , $n (G) = 1$ . Notice that $F$ and $G$ together make up all of $S$ .

Worked example 4: events with two dice

A red die and a green die are rolled. Use the grid to find the number of outcomes in each event: (a) the total is 7; (b) the total is 12; (c) both dice show the same number (a double); (d) the total is more than 9.

Solution

The shaded diagonal: $(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)$ , so 6 outcomes.
Only $(6, 6)$ : 1 outcome.
$(1, 1), (2, 2), \ldots, (6, 6)$ : 6 outcomes, down the other diagonal.
Totals 10, 11 and 12 appear $3 + 2 + 1 = 6$ times.

Warning

Order matters, and totals are not outcomes

$(4, 3)$ and $(3, 4)$ are different outcomes: in the first, the red die shows 4; in the second, the green die does. In the same way, HT and TH are different.
The totals 2, 3, …, 12 are 11 possible results, but they are not a list of equally likely outcomes: a total of 7 happens in 6 ways, a total of 12 in only one. Always count with the 36 ordered pairs.

Tip

Before you count anything, write $S$ and $n (S)$ at the top of your working. For two stages, check $n (S)$ with the rule $m \times n$ .

FuseSchool: Sample Spaces (3 min). Watch how a table of outcomes lists every combination when two things happen at once.

✦

Practice

Write each sample space as a set. Questions 5 to 7 involve two stages.

Write the sample space and $n (S)$ for spinning a spinner with 8 equal sectors numbered 1 to 8.
A die is rolled. List the events: (a) a multiple of 3; (b) an odd number; (c) a number greater than 6.
A letter is chosen from the word MATH. Write $S$ , and list the event "a vowel".
Which of these is an outcome, and which is an event, when a die is rolled: "the die shows 2"; "the die shows an even number"?
A coin is tossed and a die is rolled. Draw a tree diagram or a table, and find $n (S)$ .
Two coins are tossed. List the event "the two coins show different faces".
Two dice are rolled. How many outcomes have (a) a total of 8, (b) a total less than 5, (c) at least one 6?

Show answersHide answers

$S = \{1, 2, 3, 4, 5, 6, 7, 8\}$ , $n (S) = 8$ .
(a) ${3, 6}$ (b) $\{1, 3, 5\}$ (c) $\varnothing$
$S = \{M, A, T, H\}$ ; the event "a vowel" is ${A}$ .
"The die shows 2" is an outcome (it is also the event ${2}$ ); "an even number" is an event, $\{2, 4, 6\}$ .
$S = \{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6\}$ , $n(S) = 2 \times 6 = 12$ .
$\{HT, TH\}$ .
(a) 5: $(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)$ . (b) 6: totals 2, 3 and 4 appear $1 + 2 + 3 = 6$ times. (c) 11: six with the red die showing 6, six with the green die showing 6, minus $(6, 6)$ , counted twice.

Sources and credits

7.4: Tree Diagrams, Tables, and Outcomes, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Photo: A throw of two dice as in the game of hazard IMG 1705s.jpg, JonRichfield, Wikimedia Commons, licensed under CC BY-SA 4.0.
Video: Sample Spaces, FuseSchool - Global Education, embedded from YouTube.

The Rwandan examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1Heads or tails at Amahoro Stadium

2Two things at once

Heads or tails at Amahoro Stadium

Two things at once