Some Important Definations

Experiment, Outcomes and Sample Space:

An experiment is a process that, when we performed, we get results and the results has many observations. These observations are called the outcomes of the experiment. And the collection of all outcomes for an experiment is called a sample space.

Some examples of experiments, their outcomes and their sample spaces are given bellow-

Experiment	Outcomes	Sample Space
Toss a coin once	Head, Tail	S= {Head, Tail}
Roll a die once	1,2,3,4,5,6	S= {1,2,3,4,5,6}
Toss a coin twice	HH,HT,TH,TT	S={HH,HT,TH,TT}
Play lottery	Win, Loss	S= {Win, Loss}
Take a test	Pass, Fail	S= {Pass, Fail}
Select a worker	Male, Female	S= {Male, Female}

Event, Simple Event and Compound Event:

Event:

When an experiment has been performed, we say that some event has occured. So, an event is a collection of one or more of the outcomes of an experiment.

Simple Event:

An event is called simple event if it contains only one sample point. Simple event is denoted by E_i.

For example:

If a coin toss once, each of the final outcome (Head and Tail) for this experiment is a simple event.

E₁={Head}

E₂={Tail}

Compound Event:

An event is called compound event if it contains more than one sample points. Compound events are denoted by A_i.

For example:

In a die throwing experiment, the event with odd numbers and even numbers are compound event.

Outcomes= 1,2,3,4,5,6

Even, A₁= {2,4,6}

Odd, A₂= {1,3,5}

Mutually Exclusive and Mutually Non-Exclusive Events:

Mutually Exclusive Events:

Events that can not occur together are called mutually exclusive events. Such events do not have any common outcomes.

A∩B=0

For Example:

Considering tossing a coin once. This experiment has two outcomes.

Sample Space= {Head, Tail}

This outcomes are mutually exclusive because one and only one of them will occur when we toss this coin once.

Mutually Non-Exclusive Events:

Events that occur together are called mutually non-exclusive events. Such events have common outcomes.

A∩B=Something

For Example:

Conder the following events for one roll of a die.

A= an even number is observed= {2,4,6}

B= an odd number is observed= {1,3,5}

C= a number less than 5 is observed= {1,2,3,4}

Here,

A∩B={0} ; Mutually exclusive event

A∩C={2,4} ; Mutually non-exclusive event

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