In this chapter we will discuss the important concept of sample space used extensively in probability chapter.

At the end, solved problems are also given for your practice.

## What is Sample space ?

The **set containing all the possible outcome of the experiment** is called Sample space.

The set for sample space is represented by ” S “.

**For example;**

Consider the **experiment of rolling a dice.**

We know that after rolling the dice, we can get any number from 1 to 6. So the samples space is represented as follows;

S = { 1, 2, 3, 4, 5, 6 }

Now let’s move towards another example;

Consider the **experiment of throwing two dice together**.

Given below are the possible outcome;

HH = Getting heads on both coin

HT = Heads on first and tails on second

TH = Tails on first and heads on second

TT = Tails on both the coins

Hence there are four types of outcome possible.

The sample space for above experiment is expressed as follows;

S = { HH, HT, TH, TT}

I hope you understood the concept of sample space, let us now solve some problems related to the concept;

### Sample space – Solved example

**Example 01**

In a class, two boys and three girls are present. Two students are selected at random. Write the sample space for the experiment.

**Solution**

Here the experiment consist of selecting two student from the class.

Let the two boys are B1 & B2 and the three girls are G1, G2, & G3.

Given below are the possible outcomes;

B1 – B2 = Both student are boys

B1- G1 = First boy and first girl

B1-G2 = First boy and second girl

B1-G3 = First boy and third girl

B2- G1 = Second boy and first girl

B2-G2 = Second boy and second girl

B3-G3 = Second boy and third girl

G1- G2 = First and second girl

G2- G3 = Second and third girl

G1-G3 = First and third girl

So there are 10 possible outcomes.

**The sample space can be expressed as**;

S = { B1B2, B1G1, B1G2, B1G3, B2G1, B2G2, B2G3, G1G2, G2G3, G3G1 }

**Example 02**

A number is randomly selected from interval [2, 7]. Find the sample space for this experiment.

**Solution**

In this experiment, the number is randomly selected from interval [2, 7].

The possible outcome can be; 2, 3, 4, 5, 6 and 7.

**The sample space is expressed as follows;**

S = {2, 3, 4, 5, 6, 7 }

**Example 03**

Two dice are thrown simultaneously. Find the sample space for the experiment.

**Solution**

The possible outcome for the experiment is expressed as;

S = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }

**Example 04**

A coin is tossed and then dice is rolled only if the coin shows head. Write the sample space.

**Solution**

There are two possibilities;

(a) We toss the coin and get tails. In this case the dice will not be rolled.

Outcome is represented as = T

(b) We toss the coin and get heads. In this case we will roll the dice.

Possible outcomes are = (H,1), (H,2), (H,3), (H,4), (H,5), (H,6)

**The sample space is expressed as;**

S = {(T),(H,1), (H,2), (H,3), (H,4), (H,5), (H,6)}

**Example 05**

A coin is tossed twice. If the second toss results in head, a dice is rolled. Write the sample space for experiment.**Solution**

When coins is tossed twice, we get the following outcome;

(i) H, T

(ii) H, H

(iii) T, H

(iv) T, T

In second and fourth case, we get head in second toss. In this case we will roll the dice. The outcome is expressed as;

(H, H, 1), (H, H, 2) , (H, H, 3) , (H, H, 4) , (H, H, 5) , (H, H, 6)

(T, H, 1), (T, H, 2) , (T, H, 3) , (T, H, 4) , (T, H, 5) , (T, H, 6)

**Given below is the samples space of the experiment;**

S = { (H, T), (T, T)

(H, H, 1), (H, H, 2), (H, H, 3), (H, H, 4), (H, H, 5), (H, H, 6)

(T, H, 1), (T, H, 2), (T, H, 3), (T, H, 4), (T, H, 5), (T, H, 6) }