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Probabilities - investment science

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9. There are two propositions: (a) I flip a coin, If it is heads, you are paid $3; if it is tails, you are paid $0. It costs you $1 to participate in this proposition. You may do so at any level, or repeatedly, and the payoffs scale accordingly. (b) You may keep your money in your pocket (earning no interest). Here is a third proposition: (c) I flip the coin three times. If at least two of the flips are heads, you are paid $27; otherwise zero. How much is this proposition worth?

17. There is a market for bets on the outcome of a coin toss. The possible outcomes are heads, tails, and edge. There are three assets traded in that market:

Asset A pays $1 independent of the outcome
Asset B pays $1.50 for a head and $0 for tails and $ if the coin lands on its edge
Asset C pays $10 if and only if the coin lands on its edge (and $0 otherwise)

The prices of those assets are always constant and are fixed at $1. We assume that the payments to the winning bidders take place immediately after the outcome is determined.
(a) What is the implied risk-free interest rate in this market?
(b) What are the risk-neutral probabilities of each of the possible coin toss outcomes (heads, tails, edge)?

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Solution Summary

Computing probabilities and expected values.

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Quantitative Methods: Management Science and Probability and Stats

Please refer to the attached ms word file.

#1: The Willow Furniture Company produces tables. The fixed monthly cost of production is $8,000, and the variable cost per table is $65. The tables sell for $180 a piece.
a. For a monthly volume of 300 tables, determine the total cost, total revenue, and profit.
b. Determine the monthly break-even volume for the Willow Furniture Company.

#3: The Retread Tire Company recaps tires. The fixed annual cost of the recapping operation is $60,000. The variable cost of recapping a tire is $9. The company charges $25 to recap a tire.
a. For a monthly volume of $18,000 yards of denim, determine the total cost, total revenue, and profit.
b. Determine the annual break-even volume for the Rolling Creek Textile Mill.

#7: Andy Mendoza makes handcrafted dolls, which he sells at craft fairs. He is considering mass producing the dolls to sell in stores. He estimates that the initial investment for plant and equipment will be $25,000, whereas labor, material, packaging, and shipping will be about $10 per doll. If the dolls are sold for $30 each, what sales volume is necessary for Andy to break even?

#9: If the maximum operating capacity of the Rolling Creek Textile Mill described in problem 3 is 25,000 yards of denim per month, determine the break-even volume as a percentage of capacity.

#17: Andy Mendoza in problem 7 is concerned that the demand for his dolls will not exceed the break-even point. He believes he can reduce his initial investment by purchasing used sewing machines and fewer machines. This will reduce his initial investment from $25,000 to $17,000. However, it will also require his employees to work more slowly and perform more operations by hand, thus increasing variable cost from $10 to $14 per doll. Will these changes reduce his breakeven point?

#23. Consider a model in which two products, x and y, are produced. There are 100 pounds of material and 80 hours of labor available. It requires 2 pounds of material and 4 hours of labor to produce a unit of x, and 1 pound of material and 5 hours of labor to produce a unit of y. The profit for x is $30 per unit and the profit for y is $50 per unit. If we want to know how many units of x and y to produce to maximize profit, the model is

maximize Z = 30x + 50y
subject to
2x + 4y = 100
x + 5y = 80

Determine the solution to this problem and explain your answer.

#17. The Ramshead Pub sells a large quantity of beer every Saturday. From past sales records the pub has determined the following probabilities for sales. Compute the expected number of barrels that will be sold on Saturday.

Barrels Probability
6 .10
7 .20
8 .40
9 .25
10 .05

#18. The following probabilities for grades in management science have been determined based on past records. The grades are assigned on a 4.0 scale, where an A is a 4.0, a B a 3.0, and so on. Determine the expected grade and variance for the course.

Grade Probability
A .10
B .30
C .40
D .10
F .10

29. A warehouse distributor of carpet keeps 6,000 yards of deluxe shag carpet in stock during a month. The average demand for carpet from the stores that purchase from the distributor is 4,500 yards per month with a deviation of 900 yards. What is the probability that a customer's order will not be met during a month? (This situation is referred to as a stockout.)

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