Question

Football teams have the option of trying to score either 1 or 2 extra points after a touchdown. They get 1 point by kicking the ball through the goal posts or 2 points by running of passing the ball across the goal line. For a recent year, 1-point kicks were successful 95% of the time, while 2-point attempts were successful only 33% of the time. In either case, failure means zero points. Calculate the expected values of the 1-point and 2 point attempts. Can you think of any circumstances in which a team should make a decision different from what the expected values suggest? Explain Calculate the expected value of the 1 point attempt. Enter your answer in the answer box and then click Check Answer 2 parts remaining to search

Answer 1

Discrete Expected Value Calculations:

Let X be a discrete random variable. The expected value of X, denoted E(X), is the long run average value of the random variable X. If you repeatedly sampled from the distribution of X a very large number of times, the average value of those observations would be E(X). The expected value of X is also known as the mean of X.

To compute the expected value of X, we use the following formula:

E(X)= P(X = 1) TER,

Note that, $x\epsilon R_{x}$   means that we are summing across all possible values of xx (any value the random variable X can have). RX denotes the range of X, which is all possible values of the random variable X. In other words, the expected value is the sum of all possible values of the random variable X times their probability of occurring.

We will start by finding E(X), the expected number of points gained on average for each 1-point attempt. First, we will need to determine the distribution of X. From the question information, we know that the probability of a 1-point attempt succeeding is 95%.

Since X is the number of points earned,

P(X = 1) = 0.95

Because there is a 0.95 probability of getting 1 point, there is a

1 - 0.95 = 0.05

probability of failing and getting no points is,

P(X = 0) = 0.05

Now that we know our probabilities, we can calculate the expected number of points earned. Recall that the expected value of X can be calculated as the sum of all values X can have times their probabilities of occurrence, or

E(X)= P(X = 1) TER,

Using this formula and our known probabilities, the expected value of X is,

E(X)= xP(X = 1) = 0P(X = 0) +1P(X = 1) TER,

010.05)+110.95) = 0.95

Thus, on average, a team will score 0.93 points for every 1-point attempt.

Now, we will compute E(Y), the expected number of points earned for each 2-point attempt. As we did with X, we must first determine the distribution of Y. We know that there is a 36% chance of a team scoring 2 points on a 2-point attempt, so

P(Y = 2) = 0.33

This means that there is a

probability that a team fails a 2-point attempt, in which case they get no points.

1-0.33 = 0.67

Therefore,

PY = 0) = 0.67

We now know the complete distribution of Y, so we can compute its expected value.

E(Y) = *P(Y = y) = 0 + P(Y = 0) +2* P(Y = 2) TER,

(0 * 0.67) + (2*0.33) = 0.66

Thus, on average, a team will score 0.66 points for each 2-point attempt.

Now that we have the expected values, we can compare the two strategies. From our calculations, we know that 1-point attempts yield 0.95 points on average while 2-point attempts only yield 0.66 points on average. Therefore, according to expected values, it would be a better idea to choose to go for the 1-point attempt because it will yield more points in the long run.

However, there are cases in which it would actually be better to go for the 2-point attempt. Consider a scenario when a team has just scored a touchdown, but they are behind the other team by 2 points. Also, assume that the touchdown was scored right before the end of the game, meaning that there probably won't be any more opportunities to score points. This would be a situation in which it would be better to go for a 2-point attempt. If the team went for a 1-point attempt, they would be behind no matter what. Even if they succeed in the 1-point attempt, they will still be behind 1 point, and will lose the game. However, if they went for the 2-point attempt, they have a chance to tie the game. While the chances of success are low, a successful 2-point attempt will tie the game, forcing either a draw or overtime. Therefore, a 2-point attempt gives the team a shot at winning or tying the game, while the 1-point attempt doesn't give any advantages.