Certain, impossible and uncertain events – Elementary Probability (Senior 1 Mathematics) | Ganzaa

Elementary Probability

Lessons

Senior 1 · Mathematics

Elementary Probability

5 lessons

Lesson 2 of 5

Certain, impossible and uncertain events

20 min

"A 70% chance of rain"

A weather forecast for Kigali says: "Tomorrow, 70% chance of rain." It does not promise rain, and it does not rule it out. It measures how likely rain is, on a scale everybody can read. In mathematics that measure is called a probability, and it runs from 0 for things that can never happen to 1 for things that must happen.

Definition

Probability of an event: a number from 0 to 1 that measures how likely the event is to happen.

Notation: $P (E)$ , read "the probability of $E$ ". For example, $P(\text{heads}) = \frac{1}{2}$ .

Definition

Impossible event: an event that can never happen. Its probability is 0. As a set it is empty: $E = \varnothing$ .

Certain event: an event that must happen. Its probability is 1. As a set it is the whole sample space: $E = S$ .

Uncertain event: an event that may or may not happen, so $0 \lt P(E) \lt 1$ .

Example: with one die, "rolling a 7" is impossible, "rolling a number less than 7" is certain, and "rolling a 6" is uncertain.

Definition

Uncertain events are described in words by where they sit on the scale:

unlikely: less likely to happen than not, $P(E) \lt 0.5$ ;
even chance (evens): as likely to happen as not, $P (E) = 0.5$ ;
likely: more likely to happen than not, $P(E) \gt 0.5$ .

The probability scale

Every probability lies somewhere on this line, from 0 (impossible) to 1 (certain).

Important

Every probability lies between 0 and 1

0 \le P(E) \le 1.

Why: a probability measures what share of the possible results favour the event. A share can be nothing (0) or everything (1), but never less than nothing or more than everything. So an answer like $1.2$ , $- 0.3$ or $130\%$ always means a mistake.

A probability can be written as a fraction, a decimal or a percentage. They are the same number:

Fraction	Decimal	Percentage	In words
$0$	0	$0 %$	impossible
$\frac{1}{4}$	0.25	$25 %$	unlikely
$\frac{1}{2}$	0.5	$50 %$	even chance
$\frac{7}{10}$	0.7	$70 %$	likely
$1$	1	$100\%$	certain

How likely is the pointer to stop on blue? The more of the circle is blue, the more likely it is.

Worked example 1: describing likelihood

Describe each event as impossible, unlikely, even chance, likely or certain: (a) the month after September is October; (b) rolling 0 with a die; (c) rain falls in Kigali on at least one day in April; (d) a learner chosen at random from your school was born on a Sunday; (e) a fair coin lands tails.

Solution

Certain: it always happens.
Impossible: a die has no face showing 0.
Likely (almost certain): April is Kigali's wettest month, with rain on about 14 days on average.
Unlikely: Sunday is one of 7 days, so the chance is about 1 out of 7.
Even chance: heads and tails are equally likely.

Worked example 2: placing events on the scale

8 equal sectors: 5 blue (B), 2 orange (O), 1 green (G).

The pointer is spun once. Where do these events go on the probability scale? (a) blue (b) orange (c) green (d) red (e) blue, orange or green.

Solution The 8 sectors are equal, so each has the same chance.

Blue: 5 out of 8 sectors, $\frac{5}{8} = 0.625$ . This is more than 0.5, so likely.
Orange: 2 out of 8, $\frac{2}{8} = 0.25$ : unlikely.
Green: 1 out of 8, $\frac{1}{8} = 0.125$ : unlikely, and less likely than orange.
Red: no sector is red, so $P(\text{red}) = 0$ : impossible.
Every sector is blue, orange or green, so the probability is 1: certain.

Worked example 3: reading a forecast

The forecast says there is a 70% chance of rain tomorrow. (a) Write the probability as a decimal and describe it. (b) What is the chance that it does not rain?

Solution

$70\% = 0.7$ . This is more than 0.5, so rain is likely, but not certain.
Either it rains or it does not, so the two chances make up $100\%$ : the chance of no rain is $100\% - 70\% = 30\%$ , or 0.3. (Lesson 5 turns this into a rule.)

Warning

Common mistakes

"Likely" is not "certain". An event with probability 0.9 is very likely, but it can still fail to happen.
Two outcomes do not always mean an even chance. A football match can be won or not won, but that does not make $P(\text{win}) = 0.5$ ; it depends on the teams.
A probability outside 0 to 1 (such as 1.5, or 7 out of 5) is always wrong. Check your working.

Tip

Remember the ends of the scale: 0 = never, 1 = always, and 0.5 = half the time. Every other probability is a point between them.

Maths Genie: The Probability Scale (3 min). Watch how events are placed on a line from 0 to 1, and how the words impossible, unlikely, evens, likely and certain match the numbers.

✦

Practice

Draw a probability scale from 0 to 1 for questions 3 and 4.

Say whether each event is impossible, unlikely, even chance, likely or certain: (a) rolling an even number with a die; (b) a learner in your class is 150 years old; (c) picking a red bead from a bag that holds only red beads; (d) rolling a 3 with a die.
Write each probability as a decimal and as a percentage: (a) $\frac{3}{4}$ (b) $\frac{1}{5}$ (c) $\frac{9}{10}$ .
A bag holds 6 green beads and 2 yellow beads, all the same size. One bead is taken without looking. Mark on the scale: G, "the bead is green"; Y, "the bead is yellow"; R, "the bead is red".
Mark on your scale an event that is: (a) certain; (b) very unlikely but possible.
Which of these numbers cannot be a probability? $0.95$ , $1.05$ , $\frac{3}{7}$ , $0$ , $- 0.2$ , $100\%$ .
Mutesi says: "Tomorrow it will either rain or not rain, so the probability of rain is 0.5." Explain her mistake.

Show answersHide answers

(a) even chance (b) impossible (c) certain (d) unlikely
(a) 0.75, $75 %$ (b) 0.2, $20 %$ (c) 0.9, $90 %$
G at $\frac{6}{8} = 0.75$ (likely); Y at $\frac{2}{8} = 0.25$ (unlikely); R at 0 (impossible).
For example: (a) "the sun rises tomorrow", at 1; (b) "rolling two dice and getting a total of 12" (1 out of 36), just above 0.
$1.05$ (more than 1) and $- 0.2$ (less than 0). $100\% = 1$ is allowed.
Rain and no rain are the only two results, but they are not equally likely: in July, Kigali's driest month, rain is unlikely; in April it is likely. Two possible results give a probability of 0.5 only if they are equally likely.

Sources and credits

7.5: Basic Concepts of Probability (certain and impossible events), 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.
Video: The Probability Scale, Maths Genie, embedded from YouTube.

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

1"A 70% chance of rain"

2The probability scale

"A 70% chance of rain"

The probability scale