Question

Miguel is a golfer, and he plays on the same course each week. The following table shows the probability distribution for his score on one particular hole, known as the Water Hole.
Score 3 4 5 6 7
Probability 0.15 0.40 0.25 0.15 0.05
Let the random variable X represent Miguel’s score on the Water Hole. In golf, lower scores are better.
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Question 2
(a) Calculate and interpret the expected value of X . Show your work.
The name of the Water Hole comes from the small lake that lies between the tee, where the ball is first hit, and the hole. Miguel has two approaches to hitting the ball from the tee, the short hit and the long hit. The short hit results in the ball landing before the lake. The values of X in the table are based on the short hit. The long hit, if successful, results in the ball traveling over the lake and landing on the other side. The two approaches are shown in the following diagram.

The figure presents a diagram of one hole on a golf course with a tee, a lake, and a hole. From left to right, the diagram is as follows. A line begins at a dot labeled Tee, and moves horizontally to the right. The line reaches a shape labeled Lake, located midway between the Tee and the Hole. To the right of the lake, the line begins again and moves horizontally to the right, until it reaches a dot labeled Hole. There are two curves, each shaped similar to a parabola. The upper curve, labeled Long, begins at the Tee and moves up and to the right, forming an arch, reaches its maximum height to the left of the lake, and moves down and to the right, ending on the horizontal line slightly to the right of the Lake. The lower curve, labeled Short, begins at the Tee and moves up and to the right, forming an arch, reaches its maximum height below and to the left of the Long curve’s maximum height, and then moves down and to the right, ending on the horizontal line slightly to the left of the Lake.
A potential issue with the long hit is that the ball might land in the water, which is not a good outcome. Miguel thinks that if the long hit is successful, his expected value improves to 4.2. However, if the long hit fails and the ball lands in the water, his expected value would be worse and increases to 5.4.

(b) Suppose the probability of a successful long hit is 0.4. Which approach, the short hit or the long hit, is better in terms of improving the expected value of the score? Justify your answer.

Answer 1

1) 4.55

2) Short hit

Step-by-step explanation:

1)

The table containing the score and the relative probability of each score is:

Score 3 4 5 6 7

Probability 0.15 0.40 0.25 0.15 0.05

Here we call

X = Miguel's score on the Water Hole

The expected value of a certain variable X is given by:

$E(X)=\sum x_i p_i$

where

$x_i$ are all the possible values that the variable X can take

$p_i$ is the probability that $X=x_i$

Therefore in this problem, the expected value of MIguel's score is given by:

$E(X)=3\cdot 0.15 + 4\cdot 0.40 + 5\cdot 0.25 + 6\cdot 0.15 + 7\cdot 0.05=4.55$

2)

In this problem, we call:

X = Miguel's score on the Water Hole

Here we have that:

- If the long hit is successfull, the expected value of X is

$E(X)=4.2$

- Instead, if the long hit fails, the expected value of X is

$E(X)=5.4$

Here we also know that the probability of a successfull long hit is

$p(L)=0.4$

Which means that the probabilty of an unsuccessfull long hit is

$p(L^c)=1-p(L)=1-0.4=0.6$

Therefore, the expected value of X if Miguel chooses the long hit approach is:

$E(X)=p(L)\cdot 4.2 + p(L^C)\cdot 5.4 = 0.4\cdot 4.2 + 0.6\cdot 5.4 =4.92$

In part 1) of the problem, we saw that the expected value for the short hit was instead

$E(X)=4.55$

Since the expected value for X is lower (=better) for the short hit approach, we can say that the short hit approach is better.