Friday, October 24, 2008

EVENT & SAMPLE SPACE OF A RANDOM EXPERIMENT


Event associated with a Random Experiment 

An event associated with a Random Experiment is a Situation or a result which is based on the possible outcomes of the experiment.


For example, we toss a fair and balanced dice with its six faces marked with dots from one to six.
We find one face out of the six faces at the top. We count the number of dots on this face. 

The outcome is ‘n’, if the number of dots on the top are n = 1, 2, 3, 4, 5, 6. Thus, in this random experiment, the possible outcomes are1, 2, 3, 4, 5, 6.

The following events are associated with this Random Experiment:-

Getting the number 1 on the top face.
Getting the number 2 on the top face
Getting the number 3 on the top face

Getting the number 4 on the top face
Getting the number 5 on the top face
Getting the number 6 on the top face

All these six events are called elementary events as they are associated with the occurrence of single outcome in each case. 

However there can be many compound events (which are associated with more than one outcome) associated with the Random Experiment as below:-


Getting a number ≥ 3.
Here, the four outcomes 3, 4, 5, 6 favour the event.


Getting an even number.
Here, the three outcomes 2, 4, 6 favour the event.

Many more such compound events can be stated.

Note:- Events associated with Random Experiment are denoted by capital letters 

SAMPLE  SPACE

The set of all possible outcomes of a random experiment is called the Sample Space associated with it and it is generally denoted by S.

If E1 , E2 , E3 ,……..,En are the possible outcomes (or elementary events) of a random experiment , then ,
S = { E1 , E2 , E3 ,……..,En } is the sample space associated to it. 

I l l u s t r a t i o n 1

Consider the experiment of tossing two coins together or a coin twice. In this experiment the possible outcomes are:


  • Head on first and Head on second,
  • Head on first and Tail on second,
  • Tail on first and Head on second,
  • Tail on first and Tail on second,

If we define:

HH = Getting Head on both coins,
HT = Getting Head on first and Tail on second,
TH = Getting Tail on first and Head on second,
TT = Getting Tail on both coins.

Then,
HH, HT, TH and TT are the elementary events associated to the random experiment of tossing of two coins. The sample space associated to this experiment is given by 

S = {HH, HT, TH, TT}

I l l u s t r a t i o n 2

Consider the experiment of tossing two dice together or a die tossed twice. 


If we define:

Eij = Getting a number i on the upper face of first die and the number j on the upper face of second die,
Where i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6.

Then, Eij are the elementary events associated to 
the random experiment of tossing of two dice and are generally 
denoted by (i, j)

Thus, (1, 1), (1, 2),……, (1, 6), (2, 1), (2, 2),……….…, (2, 6),
           (3, 1),……………...…., (3, 6), ………………………..………,
            .………………………………,(6, 1),…………….…...., (6, 6), 
are 36 elementary events associated to the random experiment of tossing of two dice and the sample space associated to it is given by

S =    { (1, 1), (1, 2),…........…, (1, 6),
             (2, 1), (2, 2),……….…, (2, 6),
             (3, 1),…………….....…., (3, 6),
             ………………........………………,
             …………………………........……,
             (6, 1),…………...……...., (6, 6) }.

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