Random Experiment and Sample Space — Class 12 Statistics Notes (GSEB)

Certain Events vs Random Events

“Statistically, the probability of any one of us being here is so small that the mere fact of our existence should keep us all in a state of contented dazzlement.”

— Lewis Thomas

Certain Events

1Each person taking birth will die.
2A fruit freely falling from a tree will fall on the ground.
3If the profit per item of a trader is ₹ 10, he will earn a profit of ₹ 500 by selling 50 items.
4If a person invests ₹ 1,00,000 in a nationalised bank at 7.5% annual interest, the interest received will be ₹ 7,500.
5These events are certain — we can say with certainty that they will happen.

Random Events

1Getting head on the upper side after tossing a balanced coin.
2Getting the number 3 on the upper side when a six-faced unbiased die is thrown.
3The new baby to be born will be a boy.
4An item produced in a factory is non-defective.
5What will be the total rainfall in a certain region in the current year.
6What will be the wheat production in a state in the current year.
7What will be the result of a cricket match played between the teams of two countries.

We cannot give a precise prediction for random events; we can only get an intuitive idea about possibility. The occurrence (or non-occurrence) of these events depends upon an unknown element which is called chance. Such events are called random events. ⭐ Probability is used to numerically express the possibility of these uncertain events.

Random Experiment — Definition and Characteristics

Random Experiment

The experiment which can be independently repeated under identical conditions and all its possible outcomes are known but which of the outcomes will appear cannot be predicted with certainty before conducting the experiment is called a random experiment.

Experiment 1

Toss a balanced coin. Two possible outcomes: Head – H, Tail – T (assume the coin does not stand on its edge).

HT?

Experiment 2

Throw a balanced die with six faces marked 1, 2, 3, 4, 5, 6. Note the number on the upper face.

123456?

Experiment 3

A wheel marked with 10 numbers 0, 1, 2, …, 9 with a pointer. The number against the pointer when it stops is the winning number.

0123456789?

Characteristics of a random experiment⭐ Exam question

A random experiment can be independently repeated under almost identical conditions.

All possible outcomes of the random experiment are known, but which of the outcomes will appear cannot be predicted before conducting the experiment.

The random experiment results into a certain outcome.

The listed outcomes are the only possible outcomes of the experiment — for example, we assume the coin does not stand on its edge.

Sample Space and Sample Points

Sample Space

The set of all possible outcomes of a random experiment is called a sample space of that random experiment. The sample space is generally denoted by U or S. The elements of the sample space are called sample points.

Sample spaces of our three experiments:

Coin: U = {H, T}Die: U = {1, 2, 3, 4, 5, 6}Wheel: U = {0, 1, …, 9}

Finite Sample Space

If the total number of possible outcomes in the sample space is finite, then it is called a finite sample space. (All three experiments above have finite sample spaces.)

Infinite Sample Space

If the total number of possible outcomes is infinite, then it is called an infinite sample space.

Example — life of electric bulbs

The life (L) of bulbs produced in a factory, recorded in hours, is a real number with value 0 or more: U = {L | L ≥ 0, L ∈ R}

If maximum life is assumed to be 700 hours: U = {L | 0 ≤ L ≤ 700; L ∈ R} — still an infinite sample space. ⭐

Sample Space Box— click outcomes to mark them

Toss a balanced coin (Illustration 1 base):

n(U) = 2 sample points· also written U = {T, H}

Finite vs Infinite Sample Space

Finite vs Infinite Sample Space
Aspect	Finite	Infinite
Definition	Total number of possible outcomes is finite	Total number of possible outcomes is infinite
Coin	U = {H, T} — 2 outcomes	—
Die	U = {1, 2, 3, 4, 5, 6} — 6 outcomes	—
Bulb life	—	U = {L \| L ≥ 0, L ∈ R}; U = {L \| 0 ≤ L ≤ 700; L ∈ R}
Natural numbers	—	U = {1, 2, 3, 4, …} (Illustration 5)

The mathematical definition of probability (the next topic) requires a finitesample space — this is exactly why “infinite” matters here.

Practice Checkpoint

Your turn. Work each question in your notebook — type only the final answer here.

Type 1 Q2

2 marks

A balanced coin is tossed three times. Write the sample space for the experiment.

Try the experiment:

Type 1 Q4 (Board)

⭐ Board

2 marks

A balanced six-faced die and a balanced coin are tossed together. Write the sample space for the experiment.

Try the experiment:

Throw the die — which of the 6 faces lands?

Type 1 Q3

2 marks

A balanced coin is tossed till the first head is obtained. Write the sample space and state whether it is finite or infinite.

Try the experiment:

Finish the rest of this question type → Practice Page· then return here for the next topic

Key Takeaways

A random experiment can be repeated independently under almost identical conditions; all its outcomes are known, but the outcome of a trial cannot be predicted with certainty.
The set of all possible outcomes of a random experiment is its sample space U (or S); its elements are sample points.
U = {H, T} for a coin; U = {1, 2, 3, 4, 5, 6} for a die; U = {0, 1, …, 9} for the wheel experiment.
Two coins (or one coin twice) give U = {HH, HT, TH, TT} — 2 × 2 = 4 outcomes; two dice give 36 and three dice give 6³ = 216 outcomes by the fundamental principle of counting.
A sample space is finite or infinite; U = {1, 2, 3, 4, …} (natural numbers) and the bulb-life sets are infinite sample spaces.
Events that depend on chance are called random events; probability numerically expresses the possibility of such events.