Probability

Introduction:

Probability is a measure of uncertainty of various phenomenon. This is the most important concept in modern science.. The interpretation of the word probability involves synonyms such as chance, odds, uncertainty, prevalence, risk, expectancy etc.

“We use probability when we want to make an affirmation, but are not quite sure” writes J. R. Lucas.

To understand the approaches of probability, we have to know some basic topics like random experiment, sample space, events etc.

Random experiment:

A random experiment is an experiment whose outcomes can’t be predict.

However in most cases the collection of every possible outcome of a random experiment can be listed.

Outcomes:

A possible result of a random experiment is called its outcome.

Sample space:

A sample space of a random experiment is the collection of all possible outcomes.

Each element of sample space is called sample point.  

Sample space is denoted by S.

Example 1: What is the sample space if one coin is tossed once?

Solution: Since either a coin turn up head (H) or tail (T). Then sample space is  

 

Example 2: Describe a sample space of rolling a die.

Solution: The sample space of this experiment is

 

Exercise 16.1

In each of the following experience 1 to 7, describe the sample space for the indicated experiment.

1.     A coin is tossed three times.

 Solution: Since either a coin turn up head (H) or tail (T). Then sample space is

  

 

2.     A die is thrown 2 times.

Solution:  Since a die has six face , then If a die is thrown two times then the sample space will be

 

                

                 

                 

                 

                 

 

3.     A coin is tossed four times.

Solution: If a coin is tossed 1 time the sample space is

Then the sample space of tossing one coin four times is given by

 

         

In the 1st toss we get one head (H) and one tail (T), shown in the picture.

 

4.     One coin is tossed and one die is thrown.

Solution:  The sample space of throwing one coin once is  and that of one die is .

Hence the required sample space is given by

 

Events of a sample space:

An event is a set of outcomes of an experiment to which a probability is assigned. That is a subset of the sample space.

Occurrence of an event:

In any given experiment or trial, there is a probability that either an event occurs or it does not. The probability of the occurrence of an event lies between 0 and 1.

Types of events:

Simple event:

An event is called simple event the sample space contain only one sample point.

Example: If a die is thrown then the possibility of appearing 6 is a simple event. And is given by .

Compound event:

An event is called compound event if the sample space contain more than one sample point.

Example: If a die is thrown then the possibility of odd number appearing is compound event because there are three possibilities, i.e.,

Certain event:

An event is called certain event which is sure to occur in any experiment.

The whole sample space is called sure event.

Impossible event:

An event is called impossible event if there is no chance to occurring of an event.  is called impossible event.

Equally likely event:

The events are called equally likely event if the outcomes of an experiment are equally likely to happen.

Example: If a coin is tossed then the there are equally likely to get head and tail.

Complementary event:

For any event , the non-occurrence event is called complimentary event.

Example: Let a die is thrown. Then getting an odd face and even face are complimentary event.

Mutually exclusive event:

Two events are said to be mutually exclusive events if both can’t occur at the same time.

In other ward, let  bebe are two events of a ample space . Then they are said to be mutually exclusive if , i.e., if they are disjoint.

Mutually exclusive events always have different outcomes.

S          imple events of a sample space are always mutually exclusive.

Exhaustive events:

 Let  be n events of a sample space . Then They are said to be Exhaustive events if .

Notation of some events:

Description of the event

Theoretic notation

Not

  or

 and

 but not

Neither  nor

At least one of ,  or

All three of  and

 

Exercise 16.2

1.   A die is rolled. Let  be the event “die shows 4” and  be the event “die shows even number”. Are  and  are mutually exclusive?

Solution: Here

  The event  and

Then . Hence they are not mutually exclusive.

 

2.     A die is thrown. Describe the following events.

Solution: If a die is thrown then the sample events are

(i)               A: a number less than 7

Solution: Here

(ii)            B: a number greater than 7

Solution: Since in throwing a die there is no number occur which is greater than 7. Then

 .

(iii)         C: a multiple of 3

Solution: Here in  only multiple of 3 are 3 and 6.

Hence

(iv)          D: a number less than 4.

Solution: The number less than 4 in sample space are 1, 2, 3

Hence

(v)            E: an even number greater than 4.

Solution: Here

(vi)          F: a number not less than 3.

Solution: The number not less than 3 are 3, 4, 5, 6.

Hence  

Now

 ,

 

 

 

      

      

 

 

 

Probability function:

Let  be sample space associated to a random space. And let  be any event. Then the function P which assigns every  to a unique non-negative real number P is called probability function if it satisfies the following axioms:

(i)         P

(ii)      P, i.e.,  

(iii)     For any event , P, the number P is called probability of eliminatory event

Probability of an event:

Let  be the number of eliminatory events associated with a random experiment and  be the number of favorable cases to an event . Then the probability of occurrence of  is denoted by P and is defined by P

Some Proves of related results:

1.    P

Proof: We know that if  be the number of eliminatory events associated with a random experiment and  be the number of favorable cases to an event . Then the probability of occurrence of  is denoted by P and is defined by P

Now, since

  Implies 

  Implies   P (proved)

2.  

Proof: Let  be the number of eliminatory events associated with a random experiment and  be the number of favorable cases to an event . If  be the complement of the event  then occurrence of  is

Therefore

                         

                         

           i.e.,

Theorem 1: (Addition theorem for two events) If  and  are two events associated with a random experiment, Then .

Proof:

Let  be the sample space associated with the random experiment experiment. Also let  contains  mutually exclusive events and  be the number of eliminatory events favorable to and respectively.

Then ,  and

Now the number of events favorable to  only is , favourable to  only is  and then favourable to or i.e.,

 Number of events favorable to  is number of events favorable to  number of events favorable to  number of events favorable to .

i.e., number of events favorable to

Hence

     i.e.,

     i.e.,  (proved)

Theorem 2: (Addition theorem for three events) Let  are three events associated with a random experiment. Then

Proof: Let

Then

Putting  in the above equation,

                      

                      

                      

Corollary:

If  are mutually exclusive events, then

.

Hence

And

Theorem: For any two events  and ,

Proof: Since  then

Also  the

Since we know that,

Hence

Combining (1), (2) and (3) we get,

  (Proved)

Some related properties of Probability:

(i)       

(ii)      

(iii)    

(iv)     

Example:  and  are two events such that  and . Find

(i)          

(ii)       

(iii)    

(iv)    

Solution:

(i)       

                

                

(ii)       

                  

                  

(iii)    

                 

(iv)