stats!
posted by Peter on .
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.
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)

Whew.
Excel spreadsheets are very helpful for these kinds of calculations.
For each coach, build a probability tree. A coach can have 3 possible outcomes in each of 2 seasons, for a total of 3^2 = 9 total. The sum of the 9 possibilities should add up to 1.
Then with each possible outcome, calculate the payoff in terms of total contributions; the Expected payoff is the sum of the 9 probabilities times the payoff from each outcome. Subtract the cost of the coach and you are done.
Take it from here.