The Axioms - Probability

               The word "Probability" often comes into our day-to-day life referring to, "What is the probability of some random event ?" simply saying what at most chance the event to occur. 

  
           In my last blog "What is the probability you will open this blog ?" I told you, why for a long time, I have been in a dilemma, about why probability is so useful, my counter-argument was, "Probability tells us about the most probable event to happen based on the available ones, it does not give us the guarantee that the event to be happened for sure, then why it's so overrated ?" and I give you a classical definition of probability.

           Although the classical definition easily gives us the probability, it has some ambiguity. 😢😢

           1)  It is applicable when the total number of events are finite.

           

       2) All the events in the event space should be equally likely (they should be symmetrical with each other).

           3) It can not solve complex problems.

       To solve this problem in 1920 von Mises came up with a new definition of probability called "Frequency Definition". 

          Let E be a random experiment be repeated under uniform conditions N times and an event A occurs N(A) times then the frequency ratio is 

  

                            f(A) = N(A)/N    and     P(A) = lim f(A) for n → ∞
         

Although it solved the problem of the infinite event case it is also weak enough to form a formal definition.

                 In 1933, all the defects were encountered by Kolmogoroff in his Axiomatic Theory of Probability. He tried to analyze the problem and noticed some of the similarities,

      1) For any event either it will not happen, the chance of happening so the probability is always between 0 and 1.

          2) The probability of all events to be happen is 1.

         3) Probabilities are linearly distributed.

         4) Probability of something not happening = 1 - probability of be happening.

         He simply took some treated them as the fundamental properties of Probability and developed a theory based on the axioms. 😎😎

         Let E be a random experiment, if S is the event space and A is any event connected with E. Then,

           1) P(A) ≥ 0

           2) P(S) = 1

           3) If A, B, C, ..... are a finite or infinite sequence of pairwise mutually exclusive events, then P( A + B + C + ...) = P(A) + P(B) + P(C) +.....

These 3 can be used to define the other many properties.

Eventually, The Axioms becomes the first choice to work with probability.

            But probability deals in very simple cases if you remember I gave you the example last time, what will happen if it is given one of the events happened.

             

     This kind of problem is solved by a method called "Conditional Probability".

So The Conditional Probability of an event B if another event A has already occurred is,  P(B | A) = P(AB)/P(A) where P(A) ≠ 0.

Q. What will happen if we do many events simultaneously?

This kind of experiment is called the Compound experiment. Several methods are there to solve this all.