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OPMT 1197 Business Statistics

Lecture 6: Random Variables 1. A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cold that day, she expects to make a $5,000 profit. If it is very cold, she expects to suffer a $12,000 loss. Based upon historical data, the weather office has estimated the chances of a warm day to be 0.60; the chances of a cool day to be 0.25.

(a) Construct a probability distribution. Weather

p(x) = probability that x occurs

X = Profit (loss)

p(x)

x∙p(x)

1.00 (b)

What is the producer’s expected profit (or loss)?

(c)

Calculate the standard deviation for the profit.

(d)

What is the probability that she will make a profit?

2. A construction company is considering submitting a bid on a contract that is worth $900,000. The research and planning needed to write up the bid will cost $40,000. If they are awarded the contract, they will incur an additional $600,000 in expenses to perform the work specified in the contract. Would you advise the company to spend the $40,000 if the bid has only a 20% chance of being accepted? Use the expected profit (loss) to determine why or why not. 3. You just got offered a job selling advertising for Alive Magazine. You were offered the choice of two different compensation plans. Option 1: A salary of $75,000. Option 2: A base salary of $50,000 plus a bonus based on your sales. (see below) If your sales reach $1,200,000 you receive a $60,000 bonus; if your sales reach $1,000,000 but below $1,200,000, you receive a $20,000 bonus and if your sales are below $1,000,000 you get no bonus. You estimate there is a 40% chance of reaching $1,200,000 in sales and a 35% chance your sales will be between $1,000,000 and $1,200,000. (a) Calculate the expected value. Which compensation plan should you choose? (b)

If you were to choose option 2, what is the probability you will end up worse off?

Sol: 1. (b) On average, earn a profit of $11,450 in the LONG RUN (c) $11,694

(d) 0.85

2.

Yes, since the expected profit is $20,000 > $0 profit from not bidding.

3.

(a) $81,000 > $75,000 (b) 0.60

Pg 1 of 5

OPMT 1197 Business Statistics

Lab Exercises: Textbook Reading 5.1 1. A real estate developer wants to buy one of two adjacent parcels of land located in Surrey. Both parcels of land are currently zoned as single-family residential. The City of Surrey has announced they will change the zoning on one of the two parcels of land from single-family to multi-family dwellings (i.e. to allow the construction of apartments and townhouses) but have not yet made a decision which parcel of land will get rezoned. The real estate developer can purchase Parcel A for $3,250,000. If the zoning is changed to multi-family dwellings the land will be worth $5,000,000. If the zoning stays as single-family it will be worth only $3,000,000. They can purchase Parcel B for $2,400,000. If the land is rezoned as multi-family it will be worth $4,000,000. If it stays zoned as single-family it will be worth only $2,000,000.

After careful analysis, they believe there is a 40% chance the City will choose to change the zoning on Parcel A and a 60% chance they will instead choose to rezone Parcel B. Calculate the expected profit (or loss) for each property. Which property should they buy? 2. ABC Insurance Company issues a policy on a small boat under the following conditions: $5,000 will be paid for a total loss. If it is not a total loss, but the damage is more than $2,000, then $1,500 will be paid. Nothing will be paid for damage costing $2,000 or less and of course nothing is paid out if there is no damage. The company estimates the probability of the first three events as 0.02, 0.10, and 0.30, respectively. How much should ABC charge if they want to make a profit of $75 above the expected amount paid out in a year? 3. You work as a commercial loan officer for Western Bank. Western made a commercial loan to Kenora Construction that still has an outstanding balance of $4,000,000. The construction business is not doing well and they are not able to pay the interest on their loan. Western has to decide whether to foreclose on the loan or to work out a new payment schedule. If Western forecloses on Kenora’s loan, it will only recover $2,100,000. If Western works out a new payment schedule, and Kenora succeeds, then the bank will be repaid the entire $4,000,000. If Kenora fails and defaults on the loan, then the bank will only recover $250,000. Western estimates that the probability that Kenora will succeed is 40%. Based on the higher expected value advise Western Bank which alternative they should choose. 4. You just got offered a job selling advertising for Alive Magazine. You were offered the choice of two different compensation plans. Option 1: A salary of $75,000. Option 2: A base salary of $50,000 plus a bonus based on your sales. (see below) If your sales reach $1,200,000 you receive a $60,000 bonus; if your sales reach $1,000,000 but below $1,200,000, you receive a $20,000 bonus and if your sales are below $1,000,000 you get no bonus. You estimate there is a 40% chance of reaching $1,200,000 in sales and a 35% chance your sales will be between $1,000,000 and $1,200,000. (c)

Calculate the expected value. Which compensation plan should you choose?

(d)

If you were to choose option 2, what is the probability you will end up worse off? Pg 2 of 5

OPMT 1197 Business Statistics

5. Translink has announced the Canada Line will be extended to Steveston. The train will take one of two routes No. 3 Road or Garden City. You have an opportunity to purchase some property on each of the routes. The purchase price of the No. 3 Road property is $2,700,000. If the Canada Line is built on No. 3 Road the property will be worth $4,500,000. If the Canada Line is not built on No. 3 Road it will be worth $900,000. You may purchase the property on Garden City for $1,800,000. If the Canada Line is built on that route your property will be worth $3,500,000 and if it is not built on Garden City your property will be worth $600,000. There is a 60% chance that No. 3 Road will be selected and a 40% chance the Garden City route will be chosen. Calculate the expected profit (or loss) for each alternative. Which property should you buy? 6. Joe carefully studies any potential investment. He is currently examining the possibility of investing in HBC and has broken the potential results of the investment into four possible outcomes with accompanying probabilities. The outcomes are annual returns (in dollars) on a single share of HBC that currently sells for $40. Return on investment: Probability: (a) What is the expected return of investing in a single share of HBC (in dollars)? (b) If Joe purchases stock whenever the expected rate of return exceeds 10%, will he purchase the stock? Why or why not? 7. A restaurant manager is considering a new location for her restaurant. The projected annual cash flow for the new location is: Annual Cash Flow Probability Calculate: (a) the expected cash flow 8. Fry’s Electronics plans to offer 5-year extended warranties on their 60-inch LED tvs. Fry’s believes the repair expenses that could be incurred during the next 5 years per tv are as follows: there is a 50% chance that there will be no expense, a 40% chance of a $500 expense and a 10% chance of a $1,000 expense. How much should they charge for an extended warranty to obtain an expected profit of $75 from the sale of each warranty? 9. A company makes electronic gadgets. One out of every 25 gadgets is faulty, but the company doesn't know which ones are faulty until a buyer complains. The company makes a $5 profit on the sale of any working gadget, but suffers a loss of $60 for every faulty gadget because they have to repair the unit. Calculate the expected profit (or loss) per sale. 10. Carrier has been offering a three-year warranty on all its central air conditioning units. Until this year, they had no way of determining how often an air conditioning unit broke down and required a service call from one of their authorized service dealers. A preliminary report found that 28% of air conditioners had one service call, 15% had two service calls and 5% had three service calls. The rest did not require any service. If the average cost of a service call is $175, provide an estimate of the expected average cost of a warranty for Carrier. Pg 3 of 5

OPMT 1197 Business Statistics

11. A game show contestant is offered the choice of receiving a computer worth $2,000 or spinning the prize wheel. If he spins the prize wheel he can win one of two prizes-either a vacation or a boat. He has a 30% chance of winning a $5,000 vacation and a 15% chance of winning an $8,000. In terms of expected value, which option is the contestant’s wiser choice?

12. You park your car at an M-Park parking lot near BCIT. M-Park charges $4/day for parking. They issue a $10 parking ticket if they catch you parking without paying. You never pay for parking. Since you only get caught 16% of the time you realize it is a cheaper way to park. (a)

What is the expected daily cost of not paying for parking?

(b) M-Park wants to increase the cost of a parking ticket to discourage students from parking without paying. How much must they charge for a parking ticket so that the expected daily cost of not paying for parking is the same as the $4/day parking fee? 13. CP Rail (CP) owns the land along the Arbutus corridor in Vancouver along a rail-line that has not been used in 15 years. The land is currently zoned for transportation and green space use. The City of Vancouver has offered to pay CP $20 million dollars to purchase the property. CP would like the property rezoned for commercial-residential use. With this zoning it would be worth an estimated $160 million dollars. CP’s managers are debating two different approaches to get the rezoning approved. Option1: Passive Approach – Do Nothing CP lawyers believe that, as it stands now, there is only a 10% chance the City of Vancouver will allow the rezoning. CP will incur no costs if they pursue this option. Option 2: Active Approach -Start running the trains CP is considering running trains again along this line. CP thinks that having freight trains screeching and thundering through this upscale neighborhood will horrify the residents and cause them to complain to the City. CP hopes this will achieve two things: get the City to offer to pay more and make it more likely they will allow the rezoning. They estimate there is a 20% chance the City would offer to pay $55 million instead of just $20 million, and would increase the chances of allowing the rezoning to 15%. However, it will cost CP $10 million dollars to fix the tracks and repair the lines if they pursue this option. Calculate the expected value for each alternative. Should CP start running the trains? 14. A retailer purchased the contents of a bankrupt shoe store for $100,000. The probabilities he will be able to sell the contents of the store for $125,000, $110,000, $100,000, or $90,000 are 0.40, 0.20, 0.25, and 0.15, respectively. Calculate the retailer’s expected profit. 15. A real estate speculator must choose between two investments: An investment in an apartment building promises a profit of $200,000 with probability 0.80, or a loss of $25,000 (from a high vacancy rate and declining rental income) with probability 0.20. An investment in an office building promises a profit of $300,000 with probability 0.50, or a loss of $50,000 with a probability of 0.50.

Pg 4 of 5

OPMT 1197 Business Statistics

(b) Which investment should the speculator make if she is in financial difficulties and will go bankrupt unless she can make a profit of at least $250,000 on her next investment? 16. You intend to buy a parcel of land for $3,000,000 with the intention of selling it in two years. The price you will get for it in 2 years depends on whether the land is rezoned or not. A real estate consultant estimates the probabilities and probable selling price to be: Rezoned as No change Industrial Residential (a) What is the expected selling price? (b) Should you purchase the land? Explain with calculations. (c) Compute the standard deviation of the selling price. (d) What is the probability that you will lose money on the sale of the land? (e) If you were to experience a major financial crisis and risk going bankrupt if you lose $200,000 on the sale of the land, what should you do? (f) Suppose you need to borrow the $3,000,000 to buy the land, at 6% compounded monthly, with the entire loan repaid in 2 years. Should you still purchase the land based on your expected selling price found in part (a)? Solutions 1. Parcel A: $550,000, Parcel B: $800,000 (b) buy Parcel B 2. $325 3. $1,750,000 < $2,100,000 →better to foreclose 4.(a) $81,000 > $75,000→ 2 (b) 0.60 5.

(a) No 3 Rd: $360,000, Garden City: $40,000 loss → choose No 3 Rd

6. (a) $3.50 (b) No, because $3.50/$40×100 = 8.75% < 10% OR No, because he expects to earn only $3.50/share < $4/share he wants (10% of $40 = $4) 7. (a) $58,500 (b) $26,556.54 8. $375 9. $2.40 11.Expected value from spinning the wheel = $2,700 > $2,000 so better to spin the wheel 12. (a) $1.60/day (b) $25/ticket 13.

Do Nothing: $34,000,000 Run Trains: $38,000,000 Better to run the trains

14.

expected revenue is $110,500, expected profit = $110,500 – 100,000 = $10,500

15. (a) Apartment: $155,000 , Office Building $125,000 → Choose apartment (b) Purchase the office building because she has a 50% chance of making a $300,000 profit 16. (a) $3,220,000 (b) Yes you should buy the land because you expect to make $220,000 (c) $712,461 (large) (d) 70% (e) 70% chance of losing $200,000 → do not buy the land (f) Must pay back $3,381,479 which exceeds the expected selling price → do not buy the land Pg 5 of 5...