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. p(x) = probability that x occurs
Weather 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
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. p(x) = probability that x occurs
Weather 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
