Probability

ICSE Class 10 · Mathematics · Chapter 25

What this chapter covers

Chance in everyday life
Random experiments
Sample space
Events
Definition of probability
Simple problems on single events
Common errors and how to avoid them

Key definitions

Probability is the branch of mathematics that measures how likely an event is to happen, using a number from 0 to 1.

Key points

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain).
A random experiment has known possible outcomes, but the result of any one trial cannot be predicted.
The sample space S is the set of all possible outcomes; n(S) is their number.
Two coins give 4 outcomes, three coins give 8, and two dice give 36 ordered pairs.
An event is a subset of the sample space; its outcomes are the favourable outcomes.
An elementary event has one outcome, a sure event has all of them, and an impossible event has none.
For equally likely outcomes, P(E) = (n(E))/(n(S)).
0 ≤ P(E) ≤ 1, P(sure event) = 1 and P(impossible event) = 0.
P(overlineE) = 1 - P(E) for every event E.
The probabilities of all the elementary events of an experiment add up to 1.
In "or" questions, subtract outcomes counted in both groups; when items are removed or added, recount n(S).

Questions answered

Define a random experiment and give one example from daily life.

A random experiment is an experiment whose possible results are known in advance but whose actual result cannot be predicted with certainty before it is performed. Example: tossing a coin to decide which team bats first.

Define the sample space and write the sample space when three coins are tossed together.

The sample space is the set of all possible outcomes of a random experiment. For three coins, S = \HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\, with 8 outcomes.

What is the difference between a sure event and an impossible event? Give the probability of each.

A sure event contains every outcome of the sample space and always happens, so its probability is 1. An impossible event contains no outcome and never happens, so its probability is 0.

Why can the probability of an event never be greater than 1?

The favourable outcomes are part of the sample space, so n(E) ≤ n(S), and hence (n(E))/(n(S)) ≤ 1.

The probability that a bulb picked at random from a lot is defective is 0.08. Find the probability that it is not defective.

1 - 0.08 = 0.92.

A die is thrown once. Find the probability of getting a number less than 5.

Favourable outcomes 1, 2, 3, 4, so P = (4)/(6) = (2)/(3).

Two coins are tossed together. Find the probability of getting at most one tail.

At most one tail means \HH, HT, TH\, so P = (3)/(4).

Two dice are thrown together. Find the probability that the sum of the numbers is 7.

The pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), so P = (6)/(36) = (1)/(6).

Past board exam questions

ICSE 2027 sample paper · 1 mark

A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the numbers 1, 2, 3……12. The probability that it will point to an odd number is:

(1)/(6)
(1)/(12)
(1)/(2)
(5)/(12)

Show answer

(1)/(2)

ICSE 2027 sample paper · 3 marks

A bag contains identical cards numbered 1 to 50. A card is selected randomly. Find the probability that the card selected, is: (a) divisible by 6. (b) not divisible by 2 and 3.

Show answer

Total outcomes = 50. (a) Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 — 8 cards. P(divisible by 6) = (8)/(50) = (4)/(25). (b) A number divisible by both 2 and 3 is divisible by 6. P(not divisible by 2 and 3) = 1 - (4)/(25) = (21)/(25). (As in the board's marking scheme. If read as "divisible by neither 2 nor 3": 50 − (25 + 16 − 8) = 17 cards, P = (17)/(50).)

ICSE 2026 board paper · 1 mark

Assertion (A): If a die is rolled, the probability of getting a number greater than 6 is (1)/(6). Reason (R): There are six possible outcomes when rolling a die, \1, 2, 3, 4, 5, 6\.

(A) is true and (R) is false.
(A) is false and (R) is true.
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true but (R) is not the correct explanation of (A).

Show answer

(A) is false and (R) is true.

ICSE 2026 sample paper · 3 marks

A box containing cards numbered between 0 and 100 are shuffled and a card is picked at random. Find the probability of getting a card which is: (a) divisible by 6. (b) not divisible by 6.

Show answer

Cards numbered between 0 and 100: 1, 2, …, 99, so 99 outcomes. (a) Multiples of 6: 6, 12, 18, …, 96, i.e. 16 numbers. P(divisible by 6) = (16)/(99). (b) P(not divisible by 6) = 1 - (16)/(99) = (83)/(99).

ICSE 2026 board paper · 3 marks

A survey was conducted on 300 families having 2 children each. The results obtained are given below. | Number of girl child | 2 | 1 | 0 | Total | |---|---|---|---|---| | Number of families | 95 | 165 | 40 | 300 | If one family is selected at random, find the probability that it will have: (a) one girl child (b) one or more girl child (c) no boy child

Show answer

(a) P(one girl child) = (165)/(300) = (11)/(20). (b) P(one or more girl child) = (165 + 95)/(300) = (260)/(300) = (13)/(15). (c) No boy child means both children are girls: P = (95)/(300) = (19)/(60).

ICSE 2025 board paper · 1 mark

Which of the following cannot be the probability of any event?

(5)/(4)
0.25
(1)/(33)
67%

Show answer

(5)/(4)

ICSE 2025 sample paper · 3 marks

In a TV show, a contestant opts for video call a friend life line to get an answer from three of his friends, named Amar, Akbar & Anthony. The question which he asks from one of his friends has four options. Find the probability that: (a) Akbar is chosen for the call. (b) Akbar couldn't give the correct answer.

Show answer

(a) One of the 3 friends is chosen, each equally likely: P(Akbar is chosen) = (1)/(3). (b) The question has 4 options, only 1 of which is correct: P(correct answer) = (1)/(4), so P(Akbar couldn't give the correct answer) = 1 - (1)/(4) = (3)/(4).

ICSE 2025 compartment paper · 3 marks

There are some red, green and white marbles in a box. One marble is picked up at random from this box. If the probability of picking up a red marble is (2)/(9) and that of picking up a green marble is (4)/(9) then find the: (a) probability of picking up a white marble. (b) number of green marbles, if total number of marbles is 54. (c) probability of not picking up a red marble.

Show answer

(a) P(white) = 1 - (2)/(9) - (4)/(9) = (3)/(9) = (1)/(3). (b) Number of green marbles = (4)/(9) × 54 = 24. (c) P(not red) = 1 - (2)/(9) = (7)/(9).