LOOKS FAM part 2

At Las Vegas, a man with \$20 needs \$40, but he is too embarrassed to wire his wife for more money. He decides to invest in roulette (which he doesn’t enjoy playing) and is considering two strategies: bet the \$20 on “evens” all at once and quit if he wins or loses or bet on “evens” one dollar at a time until he has won or lost \$20. Compare the merits of the strategies.

How thick should a coin be to have a 1/3 chance of landing on edge?

In a laboratory, each of a handful of thin 9-inch glass rods had one tip marked with a blue dot and the other with a red. When the laboratory assistant tripped and dropped them onto the concrete floor, many broke into three pieces. For these, what was the average length of the fragment with the blue dot?

Shuffle an ordinary deck of 52 playing cards containing four aces. Then turn up cards from the top until the first ace appears. On the average, how many cards are required to produce the first ace?

A railroad numbers its locomotives in order, 1, 2, … , N. One day you see a locomotive and its number is 60. Guess how many locomotives the company has.

You have looked at 5 locomotive and the largest number observed is 60. Again, guess how many locomotives the company has.

If a stick is broken in two at random, what is the average length of the smaller piece?

If a stick is broken in two at random, What is the average ratio of the smaller length to the larger?

A bar is broken at random in two places. Find the average size of the smallest, of the middle-sized, and the largest pieces.

A game consists of a sequence of plays; on each play either you or your opponent scores a point, you with p (less than 1/2), he with probability of 1-p. The number of plays is to be even – 2 or 4 or 6 and so on. To win the game you must get more than half the points. You know p, say 0.45, and you get a prize if you win. You get to choose in advance the number of plays. How many do you choose?

From a shuffled deck, cards are laid out on a table one at a time, face up from left to right, and then another deck is laid out so that each of its cards is beneath a card of the first deck. What is the average number of matches of the card above and the card below in repetitions of this experiment?

A typist types of letters and envelopes to "n" different persons. The letters are randomly put into envelopes. On the average, how many letters are put into their own envelopes?

The king, to test a candidate for the position of wise man, offers him a chance to marry the young lady in the court with the largest dowry. The amounts of the dowries are written on slips of paper and mixed. A slip is drawn at random, and the wise man must decide whether that is the largest dowry or not. If he decides it is, he gets the lady and her dowry if he is correct; otherwise, he gets nothing. If he decides against the amount written on the first slip, he must choose or refuse the next slip, and so on until he chooses one or else the slips are exhausted. In all, 100 attractive young ladies participate, each with a different dowry. How should the wise man make his decision?

What is the probability that the quadratic equation x^2+2bc+c=0 has real roots?

Starting from an origin 0, a particle has a 50-50 chance of moving 1 step north, or 1 step south, and also a 50-50 chance of moving 1 step east or 1 step west. After the step is taken, the move is repeated from the new position and so on indefinitely. What is the chance that the particle returns to the origin?

A particle starts at an origin 0 in three-space. Think of the origin as centered in a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight corners of the cube. Thus, at every move the particle has a 50-50 chance of moving one unit up or down, one unit east or west, and one unit north or south. If the walk continues forever, find the fraction of particles that return to the origin.

A table of infinite expanse has inscribed on it a set of parallel lines spaced 2a units apart. A needle of length 2l (smaller than 2a) is twirled and tossed on the table. What is the probability that when it comes to rest it crosses a line?

Suppose we toss a needle of length 2L (less than 1) on a grid with both horizontal and vertical rulings spaced one unit apart. What is the mean number of lines the needle crosses?

A factory renovation is needed to build a compact microwave with a new shape, which will be called the Micro Pizza Heater. The low sales-volume prediction (20,000 heaters per year) has a subjectively estimated probability of 30%. The most likely market prediction is 30,000 units sold per year. The optimistic market prediction (30,000 sold the first year, with annual increases of 5000) has a subjectively estimated probability of 10%. In all cases, the factory equipment and the market will last 5 years. The net revenue will be \$10 per microwave. What is the probability distribution for the net revenue?

The potential costs of construction (first costs) under the three states of nature were \$225,000, \$250,000, and \$295,000, with probabilities of 0.2, 0.S, and 0.3, respectively. At the beginning of Section 18.2, the expected value or mean of this distribution was calculated to be \$258,500. What is the standard deviation of the construction cost?

A new engineer plans on depositing \$250 per month in an account that pays 6% annual interest with monthly compounding. How much of a down payment for a house has the engineer accumulated after a year?

Susan works in the electrical and mechanical engineering department of MidWestTech. She is evaluating the benefits of installing a more efficient motor on one of the assembly lines. The line is producing (and operating) 8% more each year. The more efficient motor has no salvage value after a life of 5 years, costs \$3000 more installed, and will save \$800 the first year. What is the PW if MidWestTech uses an i of 10%?

Creekside Engineering wants to establish an engineering scholarship in honor of its founder, Jane Freeborn. The endowment will be placed in a savings account by the university, and the interest income will be available for the scholarship. The account's interest rate is 6%, and the scholarship amount is \$3000 per year. What is the required endowment?

More Tech Drilling has had a downturn in business, causing it to lay off one of its six engineers One of its computer workstations has been idle for 3 months. A proposal has come forward to use the workstation for process control in the plant More Tech Drilling has identified the cost to buy and operate this workstation at \$55 per hour. If the machine is idle, the cost is \$40 per hour. If the machine is used for process control, it will be worn out in 9 years. How should this proposal be evaluated?

AAA Construction is bidding on a project whose costs are divided into \$ 20,000 for start-up. 120,000 for the first year, and \$ 80,000 for the second year. If the interest rate is 1% per month, or 12.68% per year, what is the PW with annual compounding? With monthly compounding?

Heavy Metal Recovery specializes in removing transuranic elements from hazardous waste streams. It is considering a large expansion of an existing facility in eastern Washington. The expansion requires S3M in working capital. If Heavy Metal's interest rate is 12%, find the EAC for its required working capital

Metro University must choose from among three plans for a new classroom building.
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(1) Spending \$21M now will meet the needs of the next 10 years.
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(2) Spending \$13M now will meet the needs of the next 6 years. After 6 years, \$12M would be spent on another building to meet the needs of the last 4 years.
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(3) For \$15M now, the new building can be designed for expansion later at a cost of \$7M.
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If Metro U uses an interest rate of 5%, find the PW of each alternative.

North Central Power & Light has just purchased a generating unit that comes with a 2-year warranty. Once the warranty expires, repair costs are expected to average \$3500 per year. Find the EAC for the repair costs over the 15-year life of the generating unit. The interest rate is 7%.

You own some municipal bonds that were issued at 8% with semiannual interest payments. They were to be repaid in 20 years. Calculate the effective interest rate for a \$10K bond that you can now sell for \$8200. The bond matures in 14 years.

Assume that a lightbulb has an average life of 22,000 hours; one-fifth of the bulbs fail at 20,000, 21,000, 22,000, 23,000, and 24,000 hours. The facility operates every hour of the year, and the bulbs are always on. Each bulb costs \$3.50 to buy, \$1.50 to install individually, and \$25 to install in block replacement At least 60% of the bulbs must be operating for adequate light The firm's interest rate is 10%. Compare block replacement at 21,000 hours with individual replacement.

Northern Gold Mining bought a mine site with 500,000 tons of recoverable ore for \$800,000. Of this total, \$50,000 is the land's estimated value. If 40,000 tons are processed this year, what is the cost depletion?

The revenue from 5500 ounces of gold is \$1.8M, and the expenses (other than depletion) are \$1.6M. What is the percentage depletion, and should cost or percentage depletion be claimed?

Many bridges are built of steel-reinforced concrete. As long as the steel is protected from corrosion,the bridge lives are very long. The cost difference between proper maintenance and early replacement is also very large. For example, a particular bridge would cost \$200,000 now to repair or \$18M in 10 years to replace. The state highway department claims it uses an interest rate of 8%, but it has decided to replace the bridge later due to a current shortage of funds. The stage agency also anticipates some federal funding for the replacement bridge. What is the implied interest rate for choosing to replace later rather than to repair now?

Data over the last 5 years indicate that for each fatality, there are 40 nonfatal injury accidents (\$15,000 present cost each) and 300 property damage accidents (\$2000 present cost each). What is the breakeven value of a statistical life (VSL) needed to justify a highway project if i = 8% The death rate on a particular three-lane road is 8 per 100 million vehicle-miles. Adding a lane would reduce this to 5 per 100 million, and other accidents would be reduced proportionately. The lane would cost \$1.5M per mile to build, and annual maintenance would be 3% of the first cost Assume that the lane would last 40 years. The road carries 10,000 vehicles per day.

Carol and George are both engineers. In 2001, their combined income was \$105,000. How much did their income need to be in 2008 to have the same purchasing power?

Your engineering group has just finished the preliminary design of a small refinery. The estimated cost is \$19M for a capacity of 2M barrels per year. Your management has asked you for order- of- magnitude estimates for facilities with capacities of 1M and 3M barrels per year. The managers also want a comparison of cost per barrel of capacity.

BigState has instituted an incentive for hospitals to file their claims for reimbursement under the MediState system in a specific electronic format. Assume that the number of hospitals is steady at 1000 and that 5 and 70 hospitals switched in the first and second years, respectively. To plan for the upgrade of the state's computer network, you must estimate the number of hospitals that will switch in each of the next 4 years. Assume that a Gompertz curve applies.

An investor has an option to purchase a tract of land that will be worth \$10,000 in six years. If the value of the land increases at 8% each year, how much should the investor be willing to pay now for this property?

A heat pump is being considered as a replacement for an existing electric resistance furnace. Based on winter heating requirements, it has been estimated that \$600 per year will be saved through reduced electricity bills. These savings have been computed at present electricity rates. If electricity price is expected to increase at an average rate of 14% per year into the foreseeable future, how much can we justify spending now for the heat pump if the interest rate is 12% per year? Assume that the life of the heat pump is 15 years and that its salvage value at that time is negligible.

A person has made an arrangement to borrow \$1,000 now and another \$1,000 two years hence. The entire obligation is to be repaid at the end of 4 years. If the projected interest rates in years 1,2, 3, and 4 are 10%, 12%, 12%, and 14%, respectively, how much will be repaid as a lump-sum amount at the end of four years?

A piece of new equipment has been proposed by engineers to increase the productivity of a certain manual welding operation. The initial investment (frst cost) is \$25,000, and the equipment will have a salvage value of \$5,000 at the end of a study period of 5 years. Increased productivity attributable to the equipment will amount to \$8,000 per year after extra operating costs have been subtracted from the value of the additional production. Thus, it is obvious that the "do nothing” alternative has been considered in estimating these cash flows. A cash flow diagram for this equipment s given in Figure 4-2.1. If the firm's minimum attractive rate of return (before income taxes) is 20% per year, is this proposal a sound one? Use the present worth method

A certain US. Treasury bond that matures in 8 years has a face value of \$10,000. This means that the bondholder will receive \$10,000 cash when the bond's maturity date is reached. The bond stipulates a fixed interest rate of 8% per year, but interest payments are made to the bondholder every 3 months and amount to 2% of the face value. A prospective buyer of this bond would like to earn 10% nominal interest on his investment because interest rates in the economy have risen since the bond was issued. How much should this buyer be willing to pay for the bond?

The Fly-by-Night finance company advertises a "bargain 6% plan" for financing the purchase of automobiles. To the amount of the loan being financed, 6% is added for each year money is owed. This total is then divided by the number of months over which the payments are to be made, and the result is the amount of the monthly payments. For example, a woman purchases a \$10,000 automobile under this plan and makes an initial cash payment of \$2,500. She wishes to pay the balance in 24 monthly payments

Purchase price =10,000 - Initial payment 2,500 = Balance due, 7,500 + 6% finance charge = 006 x 2
years x 7,500 =900 =Total to be paid =8,400 Monthly payments = 8,400/24 =350

What rate of interest does she actually pay?

A small company needs to borrow \$160,000. The local (and only) banker makes this statement. "We can loan you \$160,000 at a very favorable rate of 12% per year for a 5-year loan. However, to secure this loan you must agree to establish a checking account (with no interest) in which the minimum average balance is \$32,000. In addition, your interest payments are due at the end of each year and the principal will be repaid in a lump-sum amount at the end of year 5." What is the true effective interest rate being charged?

In 1990 a company generates \$1,500,000 of gross income and incurs operating expenses of \$800,000. Interest payments on borrowed capital amount to \$18,000. The total depreciation deductions in 1990 equal \$114,000. What is the taxable income of the firm?