stats!! Help long question lots of reading
posted by Peter on .
The Athletic Director at a major Midwestern University, Biff Barnhart, is about to lose another night’s sleep over the school’s ailing football team. The Beagles have just finished their 3rd consecutive losing season. At a bit past 1 AM he took another call from Ian “Abe” Egger, the Director of the Foundation:
“Barnhart you’re killing me. You got to do something, and you have got to do it NOW! Every losing season costs us millions in Alumni gifts”
Universities have long been identified by their success in Division 1 athletics, and it is certainly no secret alumni donations do rise and fall with the success or failure of the football team. In fact, the annual fund drive has been scheduled to begin in March, soon after the dust settles from the football season. Abe looked at fundraising over the last decade. In the six seasons where the football team won less than 40% of the games (losing seasons) Abe was only able to raise an average of $1.2 million in contributions to the Athletic Department (adjusted to today’s dollars). In the four seasons where the football team won between 40% and 60% of the games (neutral season) Abe was able to raise an average of $3 million in contributions to the Athletic Department. Abe is confident that if the football team were to have a winning season where they won more than 60% of the games, he could raise something in the neighborhood of $6 Million.
Unbeknownst to Abe, Biff has already decided to fire the football coach, nice guy and perennial loser, Joe Noidea. The announcement is to be made after a new coach has been hired. Joe is aware of the change and has agreed to accept a position in the School of Management as an inspirational speaker.
At the recently concluded NCAA championship, Biff interviewed a dozen candidates. He has three he really likes, and could be happy with any of them: Coach Williams, Coach Neureaga, or Coach Claven.
Coach Williams, from a perennially successful program, would command a salary of $3.5 million per year. At his current school he has a record of 3814, including 4 bowl appearances. No doubt the alumni would be thrilled. Biff estimates the following: Coach Williams has an 88% probability of producing a winning season the first year, a 10% probability of a neutral season and a 2% probability of a losing season.
Coach Claven, from a “midmajor” conference, would be willing to come on board for $1.5 million per year. Given the condition of the program, Biff believes that Coach Claven has a 34% chance of producing a winning season which is better than his estimated 26% chance of having a losing season.
Coach Neureaga, more of a gamble, has been an assistant at a big school and would require a $200,000 yearly contract. With this limited experience there isn’t much of a likelihood that he could produce a winning season the first year; say 12%. If he doesn’t have a winning season he is just as likely to produce a neutral season as a losing season.
A twoyear contract would be required. No incentives are to be offered by the University. The probabilities below have proven to be pretty much standard in college football; winning begets winning and losing begets losing.
You are attempting to determine which coach would provide the highest return to the Athletic Department, based on a twoyear contract. Your model is to be constructed using the following: After a winning season, the probability of a repeat win increases. Biff’s best guess is that the probability of a second winning season would INCREASES by ½ the difference between the probability of the first winning season and 100%. Thus, a winning season by a coach who initially had a 50% chance of having a winning season would increase Biff’s estimated probability of a second winning season by 25% to 75%. A winning season by a coach who initially had a 70% chance of having a winning season would increase the probability of a second winning season by 15% to 85%. After a winning season, the probability of a losing season is slim, and Biff thinks that 1% is the appropriate measure. After a neutral season, the probabilities remain the same as they were for the first season. After a losing season, the Biff estimates that the probability of a winning season, without a coaching change is only 1%. The probability of another losing season increases by 20% over the estimated probability for the first season. (That is an additional 20%; if a coach was deemed to have a 30% probability of a losing season and then had one, the probability for the next season would be 50%)
Since the coach will be new to the team you may assume no “history” in that each of the 3 choices has no winning, neutral or losing season to follow up on. The probabilities stated for each coach’s likelihood are Biff’s best guess for the first season.
Your task: Make a recommendation to Biff, supported by statistics. (and using his estimates)

Yep, it's a long and complex question.
What have you done so far to comprehend it, do the math, and make a tentative recommendation? 
Coach williams
First winning Season88%
Second Winning Season94%
First Winning Season 88%
Second Losing Season 1%
Firstneutral 10%
SecondWin (88) Lose(10) Neutral (2)
First Lost 10%
Second win 1%
First Lost 10
Second Lost 30
Is that right for the first coach at all?