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?

(In reply to Solution (only a common calculator used) by Penny)

The transition probability from B to G is 0.03--not 0.3. (1 - 0.15 - 0.82 = 0.03).

Given the use of the wrong probability, there are also some arithmentic mistakes:
(0.8)(0.15)(0.6)(0.2) = 0.0144, not 0.0010368.
(0.15)(0.2)(0.05)(0.3) = 0.00045, not 0.0003
(0.05)(0.82)(0.82)(0.3) = 0.010086, not 0.00413526

The array should (using the correct 0.03) look like

GGGG           .8  .8  .8  .8             0.409600


GGMG           .8  .8  .15  .2            0.019200


GGBG           .8  .8  .05  .03           0.000960


GMGG           .8  .15  .2  .8            0.019200


GMMG           .8  .15  .6  .2            0.014400


GMBG           .8  .15  .2  .03           0.000720


GBGG           .8  .05  .03  .8           0.000960


GBMG           .8  .05  .15  .2           0.001200


GBBG           .8  .05  .82  .03          0.000984


MGGG           .15  .2  .8  .8            0.019200


MGMG           .15  .2  .15  .2           0.000900


MGBG           .15  .2  .05  .03          0.000045


MMGG           .15  .6  .2  .8            0.014400


MMMG           .15  .6  .6  .2            0.010800


MMBG           .15  .6  .2  .03           0.000540


MBGG           .15  .2  .03  .8           0.000720


MBMG           .15  .2  .15  .2           0.000900


MBBG           .15  .2  .82  .03          0.000738


BGGG           .05  .03  .8  .8           0.000960


BGMG           .05  .03  .15  .2          0.000045


BGBG           .05  .03  .05  .03         0.000002


BMGG           .05  .15  .2  .8           0.001200


BMMG           .05  .15  .6  .2           0.000900


BMBG           .05  .15  .2  .03          0.000045


BBGG           .05  .82  .03  .8          0.000984


BBMG           .05  .82  .15  .2          0.001230


BBBG           .05  .82  .82  .03         0.001009

------
Totalling .52184185.

Maybe your calculator needs a repair for its intermittent problems.

Posted by Charlie on 2003-12-26 22:36:38