If the above proportions represent the probability

Question 1

In the sport of archery, which involves shooting arrows at a target, scores are awarded depending upon the accuracy of the shot. Over a long period of time, a particular competitor has made scores according to the proportions in the table below.

If the above proportions represent the probability that the competitor will achieve that score with a single shot, and the shots are considered to be independent, answer the following questions.

(a) What is the expected (mean) score of the competitor with a single  shot?

(b)   What is the expected (mean) total score of the competitor  with 50 shots?

(c)   What is the standard deviation of the competitor’s  scores?

(d) Suppose that the competitor has exactly five shots. Find the probability that they will have:

I.   No bullseyes

II.     5 bullseyes

III.  At least one bullseye

IV.   At most 3 scores of 50 or better.

V.   At least four scores of 50 or better

VI.   a total of 0 points

Question 2

Consider a five-faced regular cylinder as shown in the diagram below. The faces are of equal area so that when the cylinder is rolled and comes to a stop, each face is equally likely to end up in contact with the surface of the table. The faces are numbered 1, 2, 3, 4 and 5.

The outcome of a roll is the number on the side in contact with the table. Suppose you roll two of these cylinders independently. Your score is the total on the two outcomes. The casino charges players $6.50 to

(a)  Let X represent your score after one throw of the two cylinders.  Construct a probability distribution (probability function) for X.

(b)  Suppose that the return to the player is their score (in $). Find a fair price to charge the player for this game.

(c)     Use the distribution in (a) to find  the standard deviation of the return.

(d)    Determine  the Percentage House Margin (to two decimal places).

(e)  Assume that a gambler plays this game ten times. Find the probability that they will come out ahead (win) in at least three of these games.

(f)     If a gambler plays this game ten times, what is the expected number of times the player will come out ahead(win)?