Question

# 1.A fair die is rolled once, and the number score is noted. Let the random variable...

1.A fair die is rolled once, and the number score is noted. Let the random variable X be twice this score. Define the variable Y to be zero if an odd number appears and X otherwise. By finding the probability mass function in each case, find the expectation of the following random variables:

Part a)X
Part b)Y
Part c)X+Y
Part d)XY

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2.To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to each of its customers, asking how many of the previous year's promotions prompted orders that would not have otherwise been made. The accompanying table lists the probabilities that were derived from the questionnaire, where X is the random variable representing the number of promotions that prompted orders. If we assume that overall customer behavior next year will be the same as last year, what is the expected number of promotions that each customer will take advantage of next year by ordering goods that otherwise would not be purchased?

X   0 1   2 3   4
P(X) 0.083 0.225 0.328 0.194 0.17
Expected value =
A previous analysis of historical records found that the mean value of orders for promotional goods is 24 dollars, with the company earning a gross profit of 30% on each order. Calculate the expected value of the profit contribution next year.
Expected value =
The fixed cost of conducting the four promotions is estimated to be 20000 dollars with a variable cost of 6 dollars per customer for mailing and handling costs. What is the minimum number of customers required by the company in order to cover the cost of promotions? (Round your answer to the next highest integer.)
Breakeven point =

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3.The owner of a small firm has just purchased a personal computer, which she expects will serve her for the next two years. The owner has been told that she "must" buy a surge suppressor to provide protection for her new hardware against possible surges or variations in the electrical current, which have the capacity to damage the computer. The amount of damage to the computer depends on the strength of the surge. It has been estimated that there is a 3% chance of incurring 350 dollar damage, 4% chance of incurring 300 dollar damage, and 14% chance of incurring 75 dollar damage from a surge within the next two years. An inexpensive suppressor, which would provide protection for only one surge, can be purchased. How much should the owner be willing to pay if she makes decisions on the basis of expected value?
Expected value =

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4.A charity holds a raffle in which each ticket is sold for \$25. A total of 9000 tickets are sold. They raffle one grand prize which is a BMW M3 valued at \$55000 along with 5 second prizes of Honda motorcycles valued at \$11000 each. What are the expected winnings for a single ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents).

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5.Nick's Ski Rental rents skis, boots, and poles for \$ 20 per day. The daily cost per set of skiis is \$ 7. It includes maintenance, storage, and overhead. Daily profits depend on daily demand for skis and the number of sets available. Nick knows that on a typical weekend the daily demand for skis is given in the table.

Probability 0.125 0.125 0.425 0.15 0.175
N ofCustomers 60   61 62 63 64

a) Find the expected number of customers:
b) If 60 sets of skis are available, compute Nick's expected profit:
c) If 61 sets of skis are available, compute Nick's expected profit:
d) If 62 sets of skis are available, compute Nick's expected profit:
e) If 63 sets of skis are available, compute Nick's expected profit:
f) If 64 sets of skis are available, compute Nick's expected profit:
g) How many sets of skis should Nick have ready for rental to maximize expected profit?

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6.Below is a probability model.
Outcome 0 1.5 2 3.5
Probability 0.2 0.4 0.15 0.25

Find the expected value of the probability model.

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7.Below is a partially complete probability model. Enter the probability for the final outcome.
Outcome 2.5 3 3.5 3.5
Probability. 0.2 0.3 0.15   ?

Find the expected value of the probability model.

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8.A box contains 5 red and 5 green marbles. If 6 marbles are drawn without replacement, what is the expected number of red marbles?

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9.A box contains 5 red and 5 green marbles. If 6 marbles are drawn without replacement, what is the expected number of red marbles?

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10.A raffle has a grand prize of a Alaskan cruise valued at \$10500 with a second prize of a Rocky Point vacation valued at \$1100. If each ticket costs \$2 and 8200 tickets are sold, what are the expected winnings far a ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents).

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11.A \$85000 oil detector is lowered under the sea to detect oil fields,
and it becomes detached from the ship. If the instrument is not found within
36 hours, it will crack under the pressure of the sea. It is assumed that a
SCUBA diver will find it with probability 0.72, but it costs \$ 600 to
hire each diver.

a)How many SCUBA divers should be hired in order to maximize the expected gain?

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12.To determine whether or not they have a certain disease, 72 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of 12. The blood samples of the 12 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 12 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the desease); whereas, if the test is positive each of the 12 people will also be individually tested and, in all, 13 tests will be made on this group. Assume the probability that a person has the desease is 0.03 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 72 people.