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 Depends...
By the way, in going through various scenarios in the independent-tests assumption, 7 of the desired statistics could be made to come out exactly, as in the following set of the three tables presented:
Well Soso Poorly
Bright 0.888224 0.072408 0.039369 0.360933
Avg 0.142968 0.727862 0.12917 0.301099
Low 0 0.081176 0.918824 0.337969
p(bright) 0.88162 0.095825 0.039076
p(avg) 0.11838 0.80358 0.106956
p(low) 0 0.100595 0.853968
had done well 0.8 0.15 0.049999
had done soso 0.2 0.6 0.2
had done poor 0.049999 0.15 0.8
Excel consistently made the probability of doing poorly given having done well equal to the probability of doing well having done poorly, here 5%, in order to get exact matches on the others.
(this was with a simpler, 3-ability-level, scenario than was previously given.)
Edited on December 26, 2003, 3:03 pm
Posted by Charlie
on 2003-12-26 15:02:42