MathsICSE Class 8 · Chance and Probability

Probability of an Event: Coins, Dice, Cards and Bags (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

For equally likely outcomes, $P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$. $P(E)$ lies between $0$ (impossible) and $1$ (certain), and $P(\text{not } E) = 1 - P(E)$.

Understand Probability of an Event: Coins, Dice and Bags

The CISCE curriculum covers consolidating the notion of chance in events like tossing coins and dice, and the probability of a chance event as a number between 0 and 1.

Experiment and outcomes. A random experiment has several possible outcomes and you cannot predict which will occur. Tossing a coin: ${H, T}$ . Rolling a die: ${1, 2, 3, 4, 5, 6}$ .

Event. A set of outcomes you are interested in, such as "a prime number on a die" $= {2, 3, 5}$ .

Probability. If all outcomes are equally likely,
$ $P (E) = \frac{n ( E )}{n ( S )},$ $
where $n (S)$ is the total number of outcomes (the sample space) and $n (E)$ the favourable ones.

Range and complement. $0 \leq P (E) \leq 1$ ; $P (impossible event) = 0$ ; $P (sure event) = 1$ ; $P (E) + P (not E) = 1$ .

Two coins. Sample space ${H H, H T, T H, T T}$ : four equally likely outcomes. $P (exactly one head) = \frac{2}{4} = \frac{1}{2}$ .

Cards and bags. In a bag with 5 red, 3 blue and 2 green balls, $P (green) = \frac{2}{10} = \frac{1}{5}$ . Letters of a word: count total letters, and count repeated letters separately.

Changing the contents. If balls are added or removed, recompute both the favourable and the total numbers; the equation $\frac{favourable}{total} = given probability$ finds an unknown count.

Probability of an Event: Coins, Dice and Bags: formulas to remember

Probability: $P (E) = \frac{favourable outcomes}{total outcomes}$; Equally likely outcomes.
Complement: $P (not E) = 1 - P (E)$; Often easier.
Range: $0 \leq P (E) \leq 1$; 0 impossible, 1 certain.

How to solve Probability of an Event: Coins, Dice and Bags questions

List the sample space, count favourable outcomes, and divide.

1List or count all possible outcomes.
2Identify the favourable outcomes for the event.
3Write $P (E)$ as a fraction in lowest terms.
4Use $1 - P (E)$ for "not" events.
5For unknown counts, form an equation in the unknown.

Probability of an Event: Coins, Dice and Bags in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Picking a sweet

Suppose a jar contains 8 red sweets and some blue sweets, and the chance of picking a blue one is $\frac{3}{11}$ .

How many blue sweets are there?

Try it first, then show the solutionSolution

$\frac{b}{8 + b} = \frac{3}{11} \Rightarrow 11 b = 24 + 3 b$ .
$8 b = 24$ , so $b = 3$ .

Answer: 3 blue sweets.

Memory hook: Favourable over total: put the unknown in both.

Tricks to remember Probability of an Event: Coins, Dice and Bags

Short memory aids. Say each one aloud once, then use it on a question the same day.

Count the sample space first
Write the total number of outcomes before looking for favourable ones.
Use it when: Every probability question.
Use "not"
For "at least one", subtract the probability of "none" from 1.
Use it when: Two-coin and similar questions.

Probability of an Event: Coins, Dice and Bags: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Coin toss · 1 mark · MCQBoard level

A coin is tossed once. The probability of getting a head is:
(a) 0 (b) $\frac{1}{2}$ (c) 1 (d) 2

Try it first, then show the solutionSolution

2 outcomes, 1 favourable.

Answer: (b)

Example 2: One die · 3 marks · Short answerBoard level

A die is rolled. Find the probability of getting (i) a prime number, (ii) a number greater than 4, (iii) a multiple of 3.

Try it first, then show the solutionSolution

Total outcomes $= 6$ .
(i) Primes $2, 3, 5$ : $\frac{3}{6} = \frac{1}{2}$ .
(ii) $5, 6$ : $\frac{2}{6} = \frac{1}{3}$ . (iii) $3, 6$ : $\frac{2}{6} = \frac{1}{3}$ .

Answer: $\frac{1}{2}, \frac{1}{3}, \frac{1}{3}$

Example 3: Bag of balls · 3 marks · Short answerHarder board level

A bag has 5 red, 3 blue and 2 green balls. One ball is drawn at random. Find $P (green)$ , $P (not green)$ and $P (blue or green)$ .

Try it first, then show the solutionSolution

Total $= 10$ .
$P (green) = \frac{2}{10} = \frac{1}{5}$ .
$P (not green) = 1 - \frac{1}{5} = \frac{4}{5}$ . $P (blue or green) = \frac{3 + 2}{10} = \frac{1}{2}$ .

Answer: $\frac{1}{5}, \frac{4}{5}, \frac{1}{2}$

Example 4: Two coins · 4 marks · Short answerHarder board level

Two coins are tossed together. List the outcomes and find the probability of (i) exactly one head, (ii) at least one head, (iii) no head.

Try it first, then show the solutionSolution

Outcomes: $H H, H T, T H, T T$ (4).
(i) $H T, T H$ : $\frac{2}{4} = \frac{1}{2}$ .
(ii) $H H, H T, T H$ : $\frac{3}{4}$ . (iii) $T T$ : $\frac{1}{4}$ .
Check: (ii) $= 1 -$ (iii).

Answer: $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}$

Example 5: Unknown number of balls · 4 marks · Long answerHOTS / competency

A bag has 8 red balls and some blue balls. The probability of drawing a blue ball is $\frac{3}{11}$ . (i) Find the number of blue balls. (ii) How many red balls must be added so that $P (red) = \frac{3}{4}$ ?

Try it first, then show the solutionSolution

(i) $\frac{b}{8 + b} = \frac{3}{11} \Rightarrow 11 b = 24 + 3 b \Rightarrow b = 3$ .
(ii) After adding $r$ red balls: $\frac{8 + r}{11 + r} = \frac{3}{4} \Rightarrow 32 + 4 r = 33 + 3 r \Rightarrow r = 1$ .
Check: $\frac{9}{12} = \frac{3}{4}$ .

Answer: 3 blue balls; add 1 red ball

Where students lose marks in Probability of an Event: Coins, Dice and Bags

The slips examiners see most often, and what to write instead.

Mistake: Putting favourable over unfavourable
The denominator is the total number of outcomes.
Wrong working
$P (green) = \frac{2}{8}$ (green over non-green).
Right working
$\frac{2}{10}$ .
Do this: Write the total first.
Usually costs: Answer mark.
Mistake: Counting repeated letters once
Each letter position is a separate equally likely outcome.
Wrong working
MATHEMATICS: total 8 different letters.
Right working
Total $= 11$ letters.
Do this: Count all positions, not distinct letters.
Usually costs: Answer mark.

Exam tips

Reduce probabilities to lowest terms.
Write the sample space for coins and dice.
Probability is never negative or more than 1.

Practice questions on Probability of an Event: Coins, Dice and Bags

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The probability of an impossible event is:
(a) 0 (b) $\frac{1}{2}$ (c) 1 (d) $- 1$
Hint
Cannot happen.
Answer
(a)
Q2 · 2 marks · Short answerBoard level
A die is rolled. Find the probability of (i) an odd number, (ii) a number less than 3.
Hint
Out of 6.
Answer
(i) $\frac{3}{6} = \frac{1}{2}$ . (ii) $1, 2$ : $\frac{2}{6} = \frac{1}{3}$ .
Q3 · 3 marks · Short answerHarder board level
Cards numbered 1 to 20 are placed in a box and one is drawn at random. Find the probability that it is (i) a multiple of 5, (ii) a prime number.
Hint
List the multiples and primes.
Answer
(i) $5, 10, 15, 20$ : $\frac{4}{20} = \frac{1}{5}$ . (ii) $2, 3, 5, 7, 11, 13, 17, 19$ : $\frac{8}{20} = \frac{2}{5}$ .
Q4 · 3 marks · Short answerHarder board level
A spinner has 8 equal sectors numbered 1 to 8. Find the probability that it stops at (i) a factor of 8, (ii) a perfect square.
Hint
Factors of 8 are 1, 2, 4, 8.
Answer
(i) $\frac{4}{8} = \frac{1}{2}$ . (ii) $1, 4$ : $\frac{2}{8} = \frac{1}{4}$ .
Q5 · 4 marks · Long answerHOTS / competency
A letter is chosen at random from the word MATHEMATICS. Find the probability that it is (i) a vowel, (ii) the letter M, (iii) a letter that appears exactly once in the word.
Hint
The word has 11 letters; repeated letters are M, A, T.
Answer
Total $11$ . Vowels A, A, E, I: $\frac{4}{11}$ . M appears twice: $\frac{2}{11}$ . Letters appearing once: H, E, I, C, S: $\frac{5}{11}$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.P(sure event)?Show answer
1.
2.P(not E) in terms of P(E)?Show answer
1 - P(E).
3.Number of outcomes for two coins?Show answer
4.
4.P(prime) on a die?Show answer
1/2.

Probability of an Event: Coins, Dice and Bags: frequently asked questions

Can probability be a fraction, decimal or percent?

Fractions are standard; decimals are also fine.

Are all outcomes always equally likely?

No; the formula applies only when they are, as with a fair coin or die.