Sample Space: Overview, Questions, Preparation

A sample space is the set of all possible outcomes of an experiment. It is the universal set S pertinent to a given experiment. 

The sample space for the experiment of tossing a coin twice is given by S = {HH, HT, TH, TT}. 

The sample space for drawing a card out of the deck is the set of all cards in the deck. 

Thus, the result of any random experiment can be and has to be a part of sample space. 

Probability is defined as a quantitative measure of uncertainty. It is a numerical value that conveys the strength of our belief in the occurrence of an event. We perform many activities where the result varies or may not be the same always-on repetition under identical conditions. 

For example, when a coin is tossed, a tail or a head might turn up, but we are unsure of the result. Such experiments are called random experiments. And the collection of all the possible outcomes of any random experiment is the sample space. 

What is the subset of a sample space?

An event is a subset of the sample space. For example, the event of drawing an ace from a deck is A = {Ace of Heart, Ace of Club, Ace of Diamond, Ace of Spade}. Thus, the outcome of any event is defined in the sample space. 

Representation

The sample space for any random experiment is denoted by ‘S’ and all the possible outcomes are written within {curly brackets}. The sampling place is also termed as the universal set of any event. 

Partition of sample space

A set of events E1, E2 ,...., En is said to represent a partition of a sample space S if

(a) Ei ∩ Ej = φ, i ≠ j; i, j = 1, 2, 3,......, n

(b) Ei ∪ E2∪ ... ∪ En = S, and

(c) Each Ei ≠ φ, i. e, P (Ei) > 0 for all i = 1, 2, ..., n

While preparing Maths for class 10^th, 11^th and 12^th equal importance should be given like that of other topics. For class 10^th and 11^th, probability and statistics jointly contribute 10-12 marks, and for class 12^th, 8 marks are allotted to probability. So, understanding the concept of sample space is important. 

1.List down the sample space for a coin that is tossed three times.
Solution.  A coin has two outcomes: head (H) and tail (T). When a coin is tossed three times, the total number of possible outcomes is the cube of two, which is 8. Thus, the sample space when a coin is tossed three times is given by S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

2. Write down the sample space for the following experiment: A die is thrown two times.
Solution. While throwing a die, the possible outcomes are 1, 2, 3, 4, 5, or 6. When a die is thrown twice, the sample space is S = {(x, y): x, y = 1, 2, 3, 4, 5, 6}

The number of elements in this sample space is 6 × 6 = 36, and the sample space is:
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)}

3. Describe the sample space for an experiment where a coin is tossed, and a die is thrown.
Solution. A coin has two sides: head (H) and tail (T), and a die has six faces that are numbered from 1 to 6. Thus, when a coin is tossed, and a die is thrown, the sample space is given as follows: S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Q: What is element of a sample space known as?

Q: When is an experiment called a random experiment?

Q: Is the sample point the same as the outcome of any random experiment?

Q: Consider the experiment in which a coin is tossed repeatedly until a head comes up. Write down the sample space.

Q: What are the different types of events in probability?