A football team is losing by 14 points near the end of a game. The team scores t

Question

A football team is losing by 14 points near the end of a game. The team scores two touchdowns (worth 6 points each) before the end of the game. After each touchdown, the coach must decide whether to go for 1 point with a kick (successful 99% of the time) or 2 points with a run or pass (successful 45% of the time).

If the team goes for 1 point after each touchdown, what is the probability that the team wins? loses? ties?

If the team goes for 2 points after each touchdown, what is the probability that the team wins? loses? ties?

Develop a strategy so that the coach's team has a probability of winning the game that is greater than the probability of losing. Explain your strategy and calculate the probabilities of winning and losing the game.

Explanation / Answer

Scenario 1
If the team goes for 1 point after each touchdown it can best tie the game by collecting 14 points if both kicks are successful.
So, required prob. of tie = 0.99*0.99 = 0.9801
So, prob of losing = 1-0.9801 = 0.0199

Scenario 2
If the team goes for two points are each touchdown it can win if it collects 4 points, lose if it collects 0 and tie if it collects 2 points.

Prob of winning = 0.45*0.45 = 0.2025
Prob of losing = 0.55*0.55 = 0.3025
Prob of tie = 2*0.45*0.55 = 0.495

Scenario 3
For 1 point in 1 touchdown and 2 points in other touchdown.

Prob of winning = 0.45*0.99 = 0.4455
Prob of losing = 0.01*0.55 +0.99*0.55= 0.55( Because the team loses when it collects 1 point or 0 point, which can be when it loses both or it loses 2 points)
Prob of tie = when 2 points are successful but one point is not = 0.45*0.01 = 0.0045

So, it 1st scenario team cannot win and in 2nd and 3rd scenario, prob of winning is max in scenario3 compared to scenario2. So, the team should choose scenario3 and go for one kick and one run or pass after two touchdowns

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