A teacher said that she had observed that how well a student does on a particular quiz depends on how well or poorly he or she did on the last quiz. Then she gave the following statistics:

If you did well on a quiz, there is an 80% chance you will do well on the next quiz, a 15% chance you will do so-so, and a 5% chance you will do poorly.

If you did so-so on a quiz, there is a 20% chance you will do well on the next quiz, a 60% chance you will do so-so, and a 20% chance you will do poorly.

If you did poorly on a quiz, there is a 3% chance you will do well on the next quiz, a 15% chance you will do so-so, and an 82% chance you will do poorly. The teacher then asked the following question (which she said we'd be able to answer once we had successfully completed the class):

If you did well on the first quiz, what is the probability that you will do well on the fifth quiz in the class?

This is a classic "Markov chains" problem, with a standard solution. If you write the status change matrix A

    [ 0.80 0.15 0.05 ]


A = [ 0.20 0.60 0.20 ]


    [ 0.03 0.15 0.82 ]

then the probability of doing well in the fifth quiz is the first element of the vector [1 0 0]T times A^5. (T stands for "transpose")

Posted by Federico Kereki on 2003-12-27 01:12:39