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?

Symbols: G (Good), M (Mediocre), B (Bad)

If any quiz is G, the odds for the next quiz are G(0.8), M (0.15), B (0.05);
If any quiz is M, the next one's odds are G(0.2), M (0.6), B (0.2)
If any quiz is B, the next one's odds are G(0.3), M (0.15), B (0.82)

Assuming the first quiz is G (probability=1), here are all possibilities for the next 4 quizzes that result in the 5th quiz being G:

GGGG (0.8)(0.8)(0.8)(0.8) = 0.4096
GGMG (0.8)(0.8)(0.15)(0.2) = 0.0192
GGBG (0.8)(0.8)(0.05)(0.3) = 0.0096
GMGG (0.8)(0.15)(0.2)(0.8) = 0.0192
GMMG (0.8)(0.15)(0.6)(0.2) = 0.0010368
GMBG (0.8)(0.15)(0.2)(0.3) = 0.0072
GBGG (0.8)(0.05)(0.3)(0.8) = 0.0096
GBMG (0.8)(0.05)(0.15)(0.2) = 0.0012
GBBG (0.8)(0.05)(0.82)(0.3) = 0.00984
MGGG (0.15)(0.2)(0.8)(0.8) = 0.0192
MGMG (0.15)(0.2)(0.15)(0.2) = 0.0009
MGBG (0.15)(0.2)(0.05)(0.3) = 0.0003
MMGG (0.15)(0.6)(0.2)(0.8) = 0.0144
MMMG (0.15)(0.6)(0.6)(0.2) = 0.0108
MMBG (0.15)(0.6)(0.2)(0.3) = 0.0054
MBGG (0.15)(0.2)(0.3)(0.8) = 0.0072
MBMG (0.15)(0.2)(0.15)(0.2) = 0.0009
MBBG (0.15)(0.2)(0.82)(0.3) = 0.00738
BGGG (0.05)(0.3)(0.8)(0.8) = 0.0096
BGMG (0.05)(0.3)(0.15)(0.2) = 0.00045
BGBG (0.05)(0.3)(0.05)(0.3) = 0.000225
BMGG (0.05)(0.15)(0.2)(0.8) = 0.0012
BMMG (0.05)(0.15)(0.6)(0.2) = 0.0009
BMBG (0.05)(0.15)(0.2)(0.3) = 0.00045
BBGG (0.05)(0.82)(0.3)(0.8) = 0.00984
BBMG (0.05)(0.82)(0.15)(0.2) = 0.00123
BBBG (0.05)(0.82)(0.82)(0.3) = 0.00413526

The sum of these odds is: 0.59538706

If you did well on the first quiz, the probability that you will do well on the fifth quiz in the class is 60 %

Posted by Penny on 2003-12-26 20:58:34