Find $n(S)$ for each of the following random experiments. a) From an urn containing 5 gold and 3 silver coins, 3 coins are drawn at random. b) 5 letters are to be placed into 5 envelopes such that no envelope is empty. c) 6 books of different subjects arranged on a shelf. d) 3 tickets are drawn from a box containing 20 lottery tickets.

Find n(S) for each of the following random experiments. a) From an urn containing 5 gold and 3 silver coins, 3

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n(S)

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Concept understanding — Random Experiment and Sample Space

Random Experiment and Sample Space

Start with intuition

Think about flipping a coin. Before you flip it, you know the only possible outcomes are heads or tails. But you cannot say which one will actually come up — that depends on chance. That's the core idea: a random experiment is any process where you know what could happen, but you don't know exactly what will happen in a single trial.

Another example: rolling a die. You know the result will be one of 1, 2, 3, 4, 5, or 6. But before you roll, you cannot predict the exact number. That's randomness.

The sample space is simply the set of all possible outcomes of that random experiment. For the coin, the sample space is $\{H, T\}$ . For the die, it's $\{1, 2, 3, 4, 5, 6\}$ .

The precise definition

A random experiment satisfies three conditions:

It can be repeated any number of times under essentially the same conditions.
All possible outcomes are known in advance.
The actual outcome on any given trial cannot be predicted with certainty.

The sample space (denoted by $S$ or $\Omega$ ) is the set of all possible outcomes of a random experiment. Each individual outcome is called a sample point.

Important

The sample space must be exhaustive — it must contain every possible outcome, and nothing outside it can occur.

Examples to cement the idea

Random Experiment	Sample Space
Toss a coin	$S = \{H, T\}$
Roll a die	$S = \{1, 2, 3, 4, 5, 6\}$
Toss two coins	$S = \{HH, HT, TH, TT\}$
Roll two dice	$S = \{(1,1), (1,2), \dots, (6,6)\}$ — 36 outcomes
Draw a card from a deck	$S = \{\text{A♠}, \text{K♠}, \dots, \text{2♣}\}$ — 52 outcomes

Notice that for two coins, the order matters: $HT$ and $TH$ are different outcomes because the first coin and second coin are distinct objects.

A common mistake

Students often write the sample space for two coins as $\{HH, HT, TT\}$ , leaving out $TH$ . That is wrong. The two coins are physically distinct (or the same coin tossed twice), so $HT$ and $TH$ are different outcomes. The sample space must have 4 elements, not 3.

Watch out

Never collapse distinct outcomes just because they look similar. If the experiment distinguishes between the two coins, the sample space must reflect that.

Why this matters

The sample space is the foundation of probability. Every event (a subset of the sample space) is defined in terms of it. If you get the sample space wrong, every probability you compute from it will be wrong. So always ask: What are all the things that could happen? Then list them systematically.

a) The urn has $5+3=8$ coins; choosing 3 at random (order irrelevant) gives $n(S)=\binom{8}{3}=56$ .

b) 'No envelope empty' with 5 letters into 5 envelopes forces each envelope to get exactly one letter, i.e. a bijection between letters and envelopes: $n(S)=5!=120$ .

c) 6 distinct books arranged in a row on a shelf: $n(S)=6!=720$ .

d) Choosing 3 tickets from 20 (order irrelevant): $n(S)=\binom{20}{3}=\dfrac{20\times19\times18}{6}=1140$ .

✓Final answer

a) 56 b) 120 c) 720 d) 1140.

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