Question

1. Problem 4-05 (Algorithmic)

   In American football, touchdowns are worth 6 points. After scoring a touchdown, the scoring team may subsequently attempt to score 1 or 2 additional points. Going for 1 point is virtually an assured success, while going for 2 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. The Temple coach believes that there is a 49% 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.

   1. Assume Temple will score two touchdowns and Killeen will not score. Create a decision tree for the decision of whether Temple's coach should go for a 1-point conversion or a 2-point conversion after each touchdown. The terminal nodes in the decision tree should be either WIN or LOSE for Temple.

Success (p) WIN Go for 2 6 Win OT (49%) WIN Fail (1-p) Success 3 10 Lose OT (51%) LOSE Go for 2 Go for 1 LOSE 2 Win OT (49% WIN Success (p) 11 Lose OT (51%) Go for 2 LOSE 7 Fail (1-p) 4 LOSE Go for 1 1 LOSE Success (p) WIN Go for 2 8 Fail (1-p) LOSE 5 Win OT (49%) WIN Go for 1 Lose OT (51%) LOSE IC

1. Assume that a WIN results in a value of 1.0 and LOSE results in a value of 0. Further, assume that the probability of converting a 2-point conversion is p = 38%. Should Temple's coach go for a 1-point conversion or 2-point conversion after scoring the first touchdown?
   1. 
      The Temple coach should go
      • should go
      • should not go
      for 2 points after the first touchdown.

****PLEASE SOLVE FOR THE HIGHLIGHTED PART OF THE GRAPH AND THE THE LAST QUESTION OF WHETHER "should go" or "should not go". THANK YOU.

Answer 1

Assume that a WIN results in a value of 1.0 and LOSE results in a value of 0. Further, assume that the probability of converting a 2-point conversion is p = 38%. Should Temple's coach go for a 1-point conversion or 2-point conversion after scoring the first touchdown?

For Temple to win there are three possibilities,

1)To score two touchdowns-12 points + 2 (1pt) =14 points ->draw+50% probability of winning in over time

2)To score two touchdowns-12 pts + 1 pt + 2 pt=15 points->win

3)To score two touchdowns-12 pts+2 pt=16 points->win

Lets calculate the probability of winning in each case:

1)Probability=1*1*0.5=0.5

2)Probability=1*0.38*2 =0.76

3)Probability=0.38^2 =0.1444

But if we see after the first touchdown,

Probability to win if i take a 1 point->0.5+0.38 =0.88

Probability to win if i take 2 point->0.38*(1+0.38*0.5)+0.6*(0.38*0.5)=0.5662

Hence there is a more probability to win if i take the 1 point in the first attempt.

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