Question

after a football team scores a touchdown, the teams coach must decide whether the team should attempt to kick for an extra point or attempt a two - point conversion to earn two additional points.
suppose the mountain view high school football team just scored a touchdown. the team has a 63% chance of successfully kicking an extra point and a 21% chance of successfully completing a two - point conversion. what should the coach do to maximize the teams score?
a. the coach should direct the team to attempt to kick for an extra point because the expected value of attempting a two - point conversion is greater than the expected value of attempting an extra point.
b. the coach should direct the team to attempt a two - point conversion because the expected value of attempting a two - point conversion is less than the expected value of attempting an extra point.
c. the coach should direct the team to attempt a two - point conversion because the expected value of attempting a two - point conversion is greater than the expected value of attempting an extra point.
d. the coach should direct the team to attempt to kick for an extra point because the expected value of attempting a two - point conversion is less than the expected value of attempting to kick an extra point.

Explanation:

Step1: Calculate the expected value of kicking an extra point

The extra - point is worth 1 point. The probability of success \(p_1 = 0.63\). The expected value \(E_1\) of kicking an extra point is given by the formula \(E = x\times p\), where \(x\) is the value of the outcome and \(p\) is the probability of the outcome. So, \(E_1=1\times0.63 = 0.63\)

Step2: Calculate the expected value of a two - point conversion

The two - point conversion is worth 2 points. The probability of success \(p_2=0.21\). Using the formula \(E = x\times p\), the expected value \(E_2\) of a two - point conversion is \(E_2 = 2\times0.21=0.42\)

Step3: Compare the expected values

Since \(E_1 = 0.63\) and \(E_2=0.42\), and \(0.63>0.42\)

Answer:

D. The coach should direct the team to attempt to kick for an extra point because the expected value of attempting a two - point conversion is less than the expected value of attempting to kick an extra point.