Definition of Probability

Probability is something strange and it has been defined in different manners. We can define probability in objective or subjective manner. Let us first use objective approach to define probability.

The Classical Definition of Probability:

This definition is for equally likely outcomes. If an experiment can produced N mutually exclusive and equally likely outcomes out of which n outcomes are favorable to the occurrence of event A, then the probability of Ais denoted by P\left( A \right)and is defined as the ratio \frac{n}{N}. Thus the probability of A is given by

P\left( A \right) = \frac{{{\text{Number of  outcomes favorable to }}A}}{{{\text{Number of possible outcomes}}}} =  \frac{n}{N}

This definition can be applied in a situation in which all possible outcomes and the outcomes in the events A can be counted. This definition is due to P.S. Laplace (1749 – 1827). The classical definition is also called the priori definition of probability. The word priori is from prior and is used because the definition of base on the previous knowledge that the outcomes are equally likely. When a coin is tossed, the probability of head is assumed to be \frac{1}{2}. This probability of \frac{1}{2} is based on this classical definition of probability. It is assumed that the two faces of the coin are equally likely. In practical life the people do believe that a coin will do justice when it is tossed. In the playgrounds, the participating teams toss the coin to start the match. A coin in which probability of head is assumed to be equal to the probability of tail is called a true or uniform or an unbiased coin. But it is an all assumption. The probability of a certain event is a number which lies between0 and1. If the event does not contain any outcome, it is called impossible event and its probability is zero. If the event is as big as the sample space, the probability of the event is one because

P\left( {{\text{the event}}} \right) =  \frac{{{\text{Number of outcomes in the event}}}}{{{\text{total number of  outcomes}}}} = \frac{N}{N} = 1


When probability of an event is one, it is called a “Sure” or “Certain”, event.

Criticism:
            The classical definition of probability has always been criticized for the following reasons:

  1. This definition assumes that the outcomes are equally likely. The term equally likely is almost as difficult as the word probability itself. Thus the definition uses the circular reasoning.
  2. The definition is not applicable when the numbers of outcomes are not equally likely.
  3. The definition is also not applicable when the total number of outcomes is infinite or it is difficult to count the total outcomes or the outcomes favorable to the event. It is difficult to count the fish in the ocean. Thus it is difficult to find the probability of catching a fish of some weight say more than one kilogram.

Example:

One day 20 files were presented to an income tax officer for disposal. Five files contained bogus entries. All the files were thoroughly mixed and there was no indication about bogus files. What is the probability that one file with bogus entries is selected.

Solution:
            Here all possible outcomes  = 20
            Let A be the event that the file has bogus entries.
            Thus, number of favorable outcomes  = 5
            Here we shall apply the classical definition of probability. All the 20 files are assumed to be equally likely for the purpose of selecting a file.
            Probability of selecting a file with bogus entries is written as P\left( A \right)
                       

P\left( A \right) =  \frac{{{\text{Number of outcomes favorable to }}A}}{{{\text{Number of possible  outcomes in }}S}} = \frac{{n\left( A \right)}}{{n\left( S \right)}}


                                   

 = \frac{5}{{10}} = \frac{1}{4}

Example:

A fair die is thrown. Find the probabilities that the face on the die is (1) Maximum (2) Prime (3) Multiple of 3 (4) Multiple of 7

Solution:

There are 6 possible outcomes when a die is tossed. We assumed that all the 6 faces are equally likely. The classical definition of probability is to be applied here
            The sample space is S = \left\{ {1,2,3,4,5,6} \right\},    n\left( S \right) = 6
            (1) Let A be the event that the face is maximum
            Thus,
                        A = \left\{ 6 \right\},          n\left( A \right) = 1
            Therefore, P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( S \right)}} =  \frac{1}{6}
            (2) Let B be the event that the face is maximum
            Thus,
                        B = \left\{ {2,3,5} \right\},           n\left( B \right) = 3
            Therefore, P\left( B \right) = \frac{{n\left( B \right)}}{{n\left( S \right)}} =  \frac{3}{6} = \frac{1}{2}
            (3) Let C be the event that the face is maximum
            Thus,
                        C = \left\{ {3,6} \right\}n\left( C \right) = 2
            Therefore, P\left( C \right) = \frac{{n\left( C \right)}}{{n\left( S \right)}} =  \frac{2}{6} = \frac{1}{3}
            (4) Let D be the event that the face is maximum
            Thus,
                        D = \phi ,       n\left( D \right) = 0
            Therefore, P\left( D \right) = \frac{{n\left( D \right)}}{{n\left( S \right)}} =  \frac{0}{6} = 0   (not possible)

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