Probability ICSE Class 10: Sample Space, Events and Single-Event Problems

What Probability is about

Weightage: CISCE publishes no chapter or unit weightage for ICSE Mathematics. Probability is in the 7. Probability part of the 2027 syllabus and can be asked in both Section A and Section B of the 80-mark paper.Source: CISCE ICSE Mathematics syllabus, examination year 2027

Key concepts

Random experiments and sample space

A random experiment has outcomes that cannot be predicted exactly, though all possible outcomes are known (tossing a coin, throwing a die, drawing a card).

The sample space $S$ is the set of all possible outcomes. Write it out (or count it) before anything else:

• one coin: $\{H,T\}$, 2 outcomes;
• two coins: $\{HH,HT,TH,TT\}$, 4 outcomes;
• one die: $\{1,2,3,4,5,6\}$;
• two dice: 36 ordered pairs $(1,1),(1,2),\dots ,(6,6)$;
• one card from a pack: 52 outcomes.

Worked example

Write the sample space when two coins are tossed together, and the number of outcomes.

Try it first, then show the solutionSolution

1. First coin H or T, second coin H or T.
2. $S=\{HH,HT,TH,TT\}$: 4 outcomes. HT and TH are different outcomes.

Answer: {HH, HT, TH, TT}; 4

Events and the definition of probability

An event $E$ is a set of outcomes (part of the sample space). If all outcomes are equally likely,
$P(E)=\frac{\text{number of outcomes favourable to }E}{\text{total number of outcomes}}.$
Always simplify the fraction.

Worked example

A die is thrown once. Find the probability of getting a prime number.

Try it first, then show the solutionSolution

1. $S=\{1,2,3,4,5,6\}$, 6 outcomes.
2. Primes: 2, 3, 5 (1 is not prime): 3 outcomes.
3. $P=\frac{3}{6}=\frac{1}{2}$.

Answer: 1/2

Range of probability and complementary events

• $0\le P(E)\le 1$. A probability can be written as a fraction, a decimal or a percentage (0.25 or 25%), but never more than 1 or less than 0.
• Sure event: $P=1$ (getting a number less than 7 on a die). Impossible event: $P=0$ (getting 7).
• Complement ('not $E$'): $P(\text{not }E)=1-P(E)$.

Worked example

A die is thrown. Find the probability of not getting a 6. Which of $\frac{5}{4}$, 0.25 and 67% cannot be a probability?

Try it first, then show the solutionSolution

1. $P(6)=\frac{1}{6}$, so $P(\text{not }6)=\frac{5}{6}$.
2. $\frac{5}{4}>1$, so it cannot be a probability; 0.25 and 67% $=0.67$ are valid.

Answer: 5/6; 5/4 cannot be a probability

A pack of 52 playing cards

• 4 suits of 13 cards: spades ♠ and clubs ♣ (black), hearts ♥ and diamonds ♦ (red). So 26 red and 26 black.
• Each suit: A, 2, 3, …, 10, J, Q, K.
• Face cards: J, Q, K, so 3 per suit and 12 in all.
• 4 aces, 4 kings, 4 queens, 4 jacks.

Count carefully when two descriptions overlap (e.g. 'a heart or a king' counts the king of hearts once).

Worked example

One card is drawn from a well-shuffled pack of 52. Find the probability that it is (a) a face card, (b) a red king.

Try it first, then show the solutionSolution

1. (a) $\frac{12}{52}=\frac{3}{13}$.
2. (b) Red kings: hearts and diamonds, 2 cards: $\frac{2}{52}=\frac{1}{26}$.

Answer: (a) 3/13 (b) 1/26

Numbered cards, discs and letters

List the favourable numbers explicitly; do not count in your head.

• Primes have exactly two factors: 1 is not prime; 2 is the only even prime.
• Perfect squares include 1 ($1=1^2$).
• 'Even and a multiple of 3' means a multiple of 6; 'a multiple of 3 and 5' means a multiple of 15.
• For letters of a word, every letter counts, including repeats: EXAMINATION has 11 letters.

Worked example

Cards numbered 1 to 20 are shuffled and one is drawn. Find the probability that the number is prime.

Try it first, then show the solutionSolution

1. Primes up to 20: 2, 3, 5, 7, 11, 13, 17, 19: 8 numbers.
2. $P=\frac{8}{20}=\frac{2}{5}$.

Answer: 2/5

Two coins and two dice

With two dice there are $6\times 6=36$ equally likely ordered pairs. Draw the $6\times 6$ grid of sums when you need it.

• Doublets: $(1,1),\dots ,(6,6)$: 6.
• Sum 7 is the most likely sum (6 ways).
• $(2,6)$ and $(6,2)$ are different outcomes.

With two coins, 'at least one head' means HH, HT, TH.

Worked example

Two dice are thrown together. Find the probability that the sum is 8.

Try it first, then show the solutionSolution

1. Pairs: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2): 5 outcomes.
2. $P=\frac{5}{36}$.

Answer: 5/36

Bags of balls and unknown numbers

For a bag with balls of different colours, the total is the sum of all balls. If a probability is given, set up an equation for the unknown number.

If balls are added or removed, recount both the favourable outcomes and the total.

Worked example

A bag has 5 red balls and $x$ blue balls. The probability of drawing a blue ball is $\frac{2}{3}$. Find $x$.

Try it first, then show the solutionSolution

1. $\frac{x}{x+5}=\frac{2}{3}$.
2. $3x=2x+10$, so $x=10$.

Answer: x = 10

Maths in real life: Probability

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.

Topics in this chapter

• Probability of an Event
• Playing Cards Probability
• Dice and Coins Probability

Probability formula sheet

Probability of an event

$P(E)=\frac{n(E)}{n(S)}$

Outcomes must be equally likely.

Range

$0\le P(E)\le 1$

Sure event 1; impossible event 0.

Complement

$P(\text{not }E)=1-P(E)$

Useful for 'not' and 'at least one'.

Sample space sizes

$\text{2 coins: }4,\text{2 dice: }36,\text{cards: }52$

Ordered outcomes.

Playing cards

$26\text{ red},26\text{ black},12\text{ face cards},4\text{ of each rank}$

Face cards: J, Q, K.

How the board asks Probability

The question patterns to expect, and the method that works for each.

Tricks to remember Probability

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

• Good over all

  Probability = good outcomes ÷ all outcomes.

  Use it when: Every question.

• Never above one

  Any answer above 1 (or below 0) means a counting error.

  Use it when: Self-check.

• One is square, not prime

  1 = 1², so 1 is a perfect square; 1 has only one factor, so it is not prime.

  Use it when: Numbered-card questions.

• 12 faces, 26 red, 4 of a kind

  The three numbers you need for any card question.

  Use it when: Playing-card questions.

Probability: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In the board paper this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Probability

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

• ✗Mistake: Using the wrong total

  CISCE examiners' comments, 2024: in an assertion–reason question on a die, candidates confused the whole sample space {1, …, 6} with the set of even numbers {2, 4, 6}.

  Wrong working

  P(even) = 2/3

  Right working

  P(even) = 3/6 = 1/2

  ✓Do this: Write S first, then the favourable outcomes; the denominator is n(S).

  Usually costs: 1 mark.

• ✗Mistake: Not simplifying

  CISCE examiners' comments, 2024 and 2025: candidates left answers such as 6/8 or 2/8 unsimplified.

  ✓Do this: Always reduce to lowest terms: 6/8 = 3/4.

  Usually costs: The final mark.

• ✗Mistake: Miscounting favourable outcomes

  CISCE examiners' comments, 2024: candidates listed favourable outcomes wrongly, and for letters common to two words gave 8/8 = 1 instead of 4/8 = 1/2.

  ✓Do this: Write out the favourable outcomes as a list, then count.

  Usually costs: 1 mark per part.

• ✗Mistake: 1 as prime, or 1 left out of squares

  CISCE examiners' comments, 2025: candidates did not count 1 as a perfect square and counted 1 as a prime number.

  ✓Do this: 1 = 1² is a perfect square; 1 is not prime.

  Usually costs: 1 mark.

• ✗Mistake: Rejecting a percentage

  CISCE examiners' comments, 2025: in an MCQ, candidates chose 67% as the value that cannot be a probability, not seeing that 67% = 0.67 is less than 1.

  ✓Do this: Convert percentages and decimals to compare with 0 and 1.

  Usually costs: 1 mark.

Practice questions on Probability

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

1. Q1 · 1 mark · MCQBoard level

   The probability of a sure event is:
   (a) 0
   (b) 0.5
   (c) 1
   (d) 100

   Hint

   Certain.

   Answer

   (c)

2. Q2 · 1 mark · MCQBoard level

   A die is thrown once. The probability of getting a number greater than 4 is:
   (a) $\frac{1}{6}$
   (b) $\frac{1}{3}$
   (c) $\frac{1}{2}$
   (d) $\frac{2}{3}$

   Hint

   5 and 6.

   Answer

   (b)

3. Q3 · 1 mark · MCQHarder board level

   If $P(E)=0.35$, then $P(\text{not }E)$ is:
   (a) 0.35
   (b) 0.53
   (c) 0.65
   (d) 1.35

   Hint

   1 − P(E).

   Answer

   (c)

4. Q4 · 1 mark · Assertion-reasonBoard level

   Assertion (A): The probability of drawing a red card from a well-shuffled pack of 52 cards is $\frac{1}{2}$.
   Reason (R): A pack has 26 red cards.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   26/52.

   Answer

   (c)

5. Q5 · 4 marks · Short answerHarder board level

   Discs numbered 1 to 100 are put in a box and one is drawn. Find the probability that the number is (a) a perfect square, (b) divisible by 7, (c) a two-digit number, (d) a number whose digits add up to 5.

   Hint

   List (d): 5, 14, 23, …

   Answer

   (a) 10 squares: $\frac{1}{10}$ (b) 14 multiples of 7: $\frac{7}{50}$ (c) 10 to 99: $\frac{90}{100}=\frac{9}{10}$ (d) 5, 14, 23, 32, 41, 50: $\frac{6}{100}=\frac{3}{50}$

6. Q6 · 4 marks · Short answerHarder board level

   Two dice are thrown together. Find the probability that (a) the sum is a prime number, (b) the difference of the numbers is 2.

   Hint

   Prime sums: 2, 3, 5, 7, 11.

   Answer

   (a) Ways: 1 + 2 + 4 + 6 + 2 = 15, so $\frac{15}{36}=\frac{5}{12}$. (b) (1, 3), (3, 1), (2, 4), (4, 2), (3, 5), (5, 3), (4, 6), (6, 4): $\frac{8}{36}=\frac{2}{9}$.

7. Q7 · 3 marks · Long answerBoard level

   A bag has 3 red, 5 black and 2 white balls. One ball is drawn. Find the probability that it is (a) black, (b) not white, (c) red or white.

   Hint

   Total 10.

   Answer

   (a) $\frac{1}{2}$ (b) $\frac{4}{5}$ (c) $\frac{1}{2}$

8. Q8 · 4 marks · Long answerHarder board level

   Each letter of the word PROBABILITY is written on a card and one card is drawn. Find the probability of (a) a vowel, (b) the letter B, (c) a consonant, (d) a letter that appears more than once in the word.

   Hint

   11 letters: P, R, O, B, A, B, I, L, I, T, Y.

   Answer

   (a) O, A, I, I: $\frac{4}{11}$ (b) $\frac{2}{11}$ (c) $\frac{7}{11}$ (d) B, B, I, I: $\frac{4}{11}$ (Y is treated as a consonant).

9. Q9 · 4 marks · Long answerHOTS / competency

   A box contains 12 balls, of which $x$ are white. If 6 more white balls are added, the probability of drawing a white ball doubles. Find $x$, and both probabilities.

   Hint

   (x + 6)/18 = 2 × x/12.

   Answer

   $\frac{x+6}{18}=\frac{2x}{12}$ gives $12x+72=36x$, $x=3$. Probabilities $\frac{3}{12}=\frac{1}{4}$ and $\frac{9}{18}=\frac{1}{2}$.

10. Q10 · 4 marks · Long answerHarder board level

    One card is drawn from a pack of 52. Find the probability that it is (a) neither a heart nor a king, (b) a red face card, (c) a black ace or a red queen, (d) a 3, 4, 5 or 6.

    Hint

    For (a), remove 13 hearts and 3 other kings.

    Answer

    (a) $\frac{36}{52}=\frac{9}{13}$ (b) $\frac{6}{52}=\frac{3}{26}$ (c) $\frac{4}{52}=\frac{1}{13}$ (d) $\frac{16}{52}=\frac{4}{13}$

11. Q11 · 1 mark · MCQHOTS / competency

    A non-leap year has 365 days. The probability that it has 53 Sundays is:
    (a) $\frac{1}{7}$
    (b) $\frac{2}{7}$
    (c) $\frac{1}{365}$
    (d) $\frac{53}{365}$

    Hint

    365 = 52 weeks + 1 day.

    Answer

    (a): the one extra day must be a Sunday, 1 out of 7 possibilities.

12. Q12 · 5 marks · Long answerHOTS / competency

    A piggy bank has fifty ₹1 coins, twenty ₹2 coins, ten ₹5 coins and five ₹10 coins. One coin falls out at random. Find the probability that it is (a) a ₹5 coin, (b) not a ₹10 coin, (c) worth at least ₹2. (d) Find the average value of a coin, and the probability that the coin that falls is worth more than this average.

    Hint

    85 coins; total ₹190.

    Answer

    (a) $\frac{10}{85}=\frac{2}{17}$ (b) $\frac{80}{85}=\frac{16}{17}$ (c) $\frac{35}{85}=\frac{7}{17}$ (d) Average $=\frac{190}{85}\approx ₹2.24$; coins worth more: ₹5 and ₹10, $\frac{15}{85}=\frac{3}{17}$.

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. 1.Definition of probability (equally likely outcomes)?Show answer

   Favourable outcomes ÷ total outcomes.

2. 2.Range of a probability?Show answer

   0 to 1 inclusive.

3. 3.P(not E)?Show answer

   1 − P(E)

4. 4.Outcomes when two dice are thrown?Show answer

   36

5. 5.Number of face cards in a pack?Show answer

   12

6. 6.Is 1 prime? Is 1 a perfect square?Show answer

   Not prime; yes, a perfect square.

7. 7.Probability of an impossible event?Show answer

   0

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Do you write the sample space first?
• ☐Do you list favourable outcomes rather than guess?
• ☐Do you simplify every answer?
• ☐Do you remember 1 is not prime but is a square?
• ☐Do you know the card counts?
• ☐Do you recount the total after cards or balls change?

Probability: frequently asked questions