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.
Question 1
Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.

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.

(a) Suppose one of Miguel’s scores from the Water Hole is selected at random. What is the probability that Miguel’s score on the Water Hole is at most 5 ? Show your work.

(b) 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.

(c) 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.

(d) Let p represent the probability of a successful long hit. What values of p will make the long hit better than the short hit in terms of improving the expected value of the score? Explain your reasoning.