Random experiment, Sample space, Sample point

Random experiment, Sample space, Sample point

Experiment: In Statistics, by the word experiment it means ‘an attempt to produce a result’. It need not be a laboratory experiment.

Random Experiment: If an experiment is such that

(i) all the possible outcomes of the experiment are predictable, in advance

(ii) outcome of any trial of the experiment is not known, in advance, and

(iii) it can be repeated any number of times under identical conditions, is called a random experiment.

Sample space: The set of all possible outcomes of a random experiment is called the sample space of the experiment and is usually denoted by S (or Ω). If S contains only finite number of elements, it is termed as finite sample space. If S contains countable number of elements, S may be called as countable sample space or discrete sample space. Otherwise, S is called an uncountable sample space.

Sample Point: The outcome of a random experiment is called a sample point, which is an element in S.

Example 8.1

Consider the random experiment of tossing a coin once “Head” and “Tail” are the two possible outcomes. The sample space is S = {H, T}. It is a finite sample space which is presented in fig. 8.1.

Example 8.2

Suppose that a study is conducted on all families with one or two children. The possible outcomes, in the order of births, are: boy only, girl only, boy and girl, girl and boy, both are boys and both are girls. Then, the sample space is S = {b,g,bg,gb,bb,gg}. It is also a finite sample space. Here, ‘b’ represents the child is a boy and ‘g’ represents the child is a girl.

Example 8.3

Consider the experiment of tossing a coin until head appears. Then, the sample space of this experiment is S = { (H), (T,H), (T,T,H), (T,T,T,H), … }. This is a countable sample space. If head appears in the first trial itself, then the element of S is (H); if head appears in the second attempt then the element of S is (T,H); if head appears in the third attempt then the element of S is (T,T,H) and so on.

Example 8.4

In the experiment of observing the lifetime of any animate or inanimate things, the sample space is

S = {x: x≥0},

where x denotes the lifetime. It is an example for uncountable sample space.

Event: A subset of the sample space is called an event. In this chapter, events are denoted by upper case English alphabets and the elements of the subsets by lower case English alphabets.

In Example 8.2, the event that the eldest child in the families is a girl is represented as

A = {g, gb, gg}

The event that the families have one boy is represented as

B = {b, bg}.

In Example 8.4, A refers to the event that the refrigerator works to a maximum of 5000 hours. Then A = { x: 0 < x ≤ 5000 } is the subset of

Mutually exclusive events

Two or more events are said to be mutually exclusive, when the occurrence of any one event excludes the occurrence of other event. Mutually exclusive events cannot occur simultaneously.

In particular, events A and B are said to be mutually exclusive if they are disjoint, that is, A ∩ B = Ø

Consider the case of rolling a die. Let A = {1, 2, 3} and B = {4, 5, 6} be two events. Then we find A ∩ B = Ø . Hence A and B are said to be mutually exclusive events.