I am part of a group of teachers in DFW area. We are very competitive when it comes to our profession. So we like to have a little fun throughout the year by having “test battles”. We simply compare results from our pupils to see who “wins”. A win is gained when one student’s score is better than the other on a given test. So there are different grade levels with different classes and different students from each class involved.
I was trying to figure out what the probability of a victory for each class by doing a set of steps within the following scenario.
There are about 12 classes involved. This week we will focus a one battle. Mr. White vs Mr. Black. Mr. White’s class has won 63% of its battles. Mr. Black’s class has won 71% of its battles. At first I was calculating the probability of a victory for Mr. Whites class given that Mr. Black wins 71% of his battles ( which means the he loses 29% of his battles). So I came up with 41% that Mr. White’s class wins the battle that week using Baye’s Theory. But then I started thinking… Each student has their own record or wins and losses in the battles as well. So then I thought, well I should be figuring out the probability of a win for Mr. White’s class based on the individual students’ records. So then I used that data. Mr. White’s student Jenny wins 77% of her battles and Mr. Black’s student Bobby wins 59% of his battles. So using the individual student data, Mr. White’s class has a 69% chance of winning the battle that week. I got that number by using Bayes theory and stating : The probability that Mr. White’s class wins the battle( Jenny 77%) given that Mr. Blacks chances of losing are 41%(from Bobby) is 69%.
Here is where I get lost.
Which is the best indicator of the true probability of Mr. White wining,?
A) The Class records as data which gives 41% chance of victory B) The individual student record which gives a 69% chance of victory OR C) A combination of both? I would state “ the probability of Mr. White’s class winning is 60% given that the class vs class data gives a 41% chance of winning and student vs student data gives a 69% chance of winning
From what I have read, the more relevant factors you add to Bayes theory the more accurate probability you can get.
Another issue I have with this whole thig is; can I use the class data with the student data since the students belong to the class to begin with and their individual results contribute to the class overall performance?