# UNDERSTANDING EXPERIMENTS AND SAMPLE SPACES

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In this article, we will be talking about experiments and sample spaces. Stay focus so you don’t miss any point.

An experiment is an act of making an observation. It is called a random experiment if

1. the experiment is such that it is impossible to state a particular outcome, but we can define the set of all possible outcomes. Each outcome is called a sample point.
2. the experiment may be repeated infinitely under unchanged conditions.
3. the outcomes of repeated experiments occur in a random fashion, or by chance.

The set of all possible outcomes of the experiment is called the sample space. denoted by S. A sample space may be discrete or continuous.

A discrete sample space is one that contains a finite or uncountably infinite number of sample points.

On the other hand, a continuous sample space is one that contains an uncountable number of sample points.

Several experiments and associated sample spaces are illustrated below:

1. Toss a coin once and observe and the number that shows upwards, The sample spaces
S = { H, T }
2. Toss a coin twice. The sample space is S = { HH, HT, TH, TT }
3. Toss a coin three times and observe the sequence of heads and tails. The sample space is
S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
4. Toss a die and observe the number of dots that is upwards
S = { 1, 2, 3, 4, 5, 6 }
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### Statistical Event

An event A associated with the sample space S. In our discussion in set theory it follows that both ∅ and S are subsets of S.

we shall usually denote events by capital letters. Some examples of events relating to ∈1, ∈2, ∈3, and ∈4 are listed below:

∈1: A1 : The coin tosses yields a head. A1 = { H }

∈2 : A2 : At least one head is observed. A2 = { HH, HT, TH }

∈3 : A3 : More than two tails occur. A3 = { T T T }

∈4 : A4 : An even number is observed. A4 = { 2, 4, 6 }

### Relative Frequency of an event

Let an experiment ∈ be repeated n times and let S be the sample space associated with ∈.

Also, let A and B be two events defined on S with nλ and n∅ as the number of times A and B occur respectively in the n repetitions.

The relative frequency of A.denoted fΛ.