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I KNOW WHAT YOU WILL BE DOING TOMORROW

 

     Say, you have a habit of studying in the morning of some days; spend time in gym in 'other days or play video games.
    We have probabilities for your habits.  Suppose you study one day.  There is chance of 70% or (.7) for going to gym the next day.  But going to video games has only the probability of 0.1.  But studying again next day has the probability 0.6.
    Your behavior pattern can be depicted using matrix as given below.

               study    gym    video     next day

 study        0.6     0.3       0.1
 gym         0.7    0.1       0.2
video        0.1    0.2       0.7
given day

     Consider the last element 0.7 in the matrix.  It says, "if you  play video game one day, returning to the video games next day has a high probability of 0.7.
    Consider the middle element in the last row 0.2, it tells us that if you play video one day, you may go to gym the next day, the chances are only 20% or 0.2.
     This is how, we have to interpret the matrix.  Now I can tell what you will be doing tomorrow or 3 days later or a week later.
    Let us assume that you study first day.  Let us represent it in simple matrix form.
        study, gym, game
        (1,         0,       0)

     Now, let us multiply this starting vector and the probability matrix to know what you will be doing the next day.

 (1, 0, 0)* (.6  .3  .1
               .7  .1 .2    =   (.6 .3 .1)
                .1 .2 .7)                                           

 refer matrix multiplication

    (.6 .3 .1) is the set of probabilities for next day for(study, gym, video).  You have high probabilities of studying again.  with the new probabilities we will find out what you are up to on Day 2.

 AGAIN MATRIX MULTIPLICATION:
(.6,.3,.1)*(.6 .3 .1
                 .7  .1 .2   =( .58,.23..19)
                .1   .2 .7) 

 again you have high chance for study.  Let us see what will happen third day.

 (.58,.23,.19) * (.6 .3 .1
                       .7 .1 .2  =(.528, .235, .237)
                       .1 .2 .7)

 'study fever' continues.  We can go on repeat the process and find what will happen in the future.  This process of linking is called Markov chain.  Google search engine uses 'Markov chaining'.
     The elegant matrix form and Markov chaining yields beautiful results.               
--------------------------------------------------------------------------------------------------
 FOOT NOTE:
(.6*1 +.3*0+.1*0       (.6
 .3*1+.1*0+.2*0    =  .3
 .1*1+.2*0+.7*0)       .1)
multiplying row with columns and adding is the method of matrix multiplication

    

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