# Sample space in probability

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) }