In American football, touchdowns are worth 6 points. After scoring a touchdown, the scoring team may subsequently attempt to score one or two additional points. Going for one point is virtually an assured success, while going for two points is successful only with probability p. Consider the following game situation. The Temple Wildcats are losing by 14 points to the Killeen Tigers near the end of regulation time. The only way for Temple to win (or tie) this game is to score two touchdowns while not allowing Killeen to score again. The Temple coach must decide whether to attempt a 1-point or 2-point conversion after each touchdown. If the score is tied at the end of regulation time, the game goes into overtime where the first team to score wins. The Temple coach believes that there is a 50% chance that Temple will win if the game goes into overtime. The probability of successfully converting a 1-point conversion is 1.0. The probability of successfully converting a 2-point conversion is p.

a. Assume Temple will score two touchdowns and Killeen will not score. Define the state of the system as the number of points by which Temple is currently trailing. Create a tree diagram for the situation in which Temple’s coach attempts a 2-point conversion after the first touchdown. If the 2-point conversion is successful, Temple will go for 1 point after the second touchdown to win the game. If the 2-point conversion is unsuccessful, Temple will go for 2 points after the second touchdown in an attempt to tie the game and go to overtime.

b. Create the transition probability matrix for this decision problem.

c. If Temple’s coach goes for a 1-point conversion after each touchdown, the game is assured of going to overtime and Temple will win with probability 0.5. For what values of p is the strategy defined in part a superior to going for 1 point after each touchdown?