There are four pens: Red, Green, Blue and Purple in a desk drawer of which two pens are selected at random one after the other with replacement. State the sample space and the following events. a) $A$: Selecting at least one red pen. b) $B$: Two pens of the same colour are not selected.

There are four pens: Red, Green, Blue and Purple in a desk drawer of which two pens are selected at random one

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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.

Since the two pens are drawn one after another WITH replacement, every draw is from the full set {Red, Green, Blue, Purple}, so the sample space is every ordered pair $(X,Y)$ with $X,Y\in\{R,G,B,P\}$ : $S=\{(R,R),(R,G),(R,B),(R,P),(G,R),(G,G),(G,B),(G,P),(B,R),(B,G),(B,B),(B,P),(P,R),(P,G),(P,B),(P,P)\},$ so $n(S)=16$ .

Event $A$ (at least one red pen) collects every pair containing R: $A=\{(R,R),(R,G),(R,B),(R,P),(G,R),(B,R),(P,R)\}$ , so $n(A)=7$ .

Event $B$ (two pens of the same colour are NOT selected) collects every pair with $X\ne Y$ : that is all 16 pairs except the 4 'matching' ones $(R,R),(G,G),(B,B),(P,P)$ , so $n(B)=12$ .

✓Final answer

n(S)=16

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Q.There are four pens: Red, Green, Blue and Purple in a desk drawer of which two pens are selected at random one after the other with replacement. State the sample space and the following events. a) AAA: Selecting at least one red pen. b) BBB: Two pens of the same colour are not selected.

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Q.There are four pens: Red, Green, Blue and Purple in a desk drawer of which two pens are selected at random one after the other with replacement. State the sample space and the following events. a) $A$ : Selecting at least one red pen. b) $B$ : Two pens of the same colour are not selected.

Unlock everything free for 14 days