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Quantitative Methods
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QMS 102
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Quantitative Methods

QMS 102

Jason Chin- Tiong Chan

Fall

Description

Business Statistics I-QMS102 Chapter5and 6
Probability
The chance that an event will occur
1. Discuss the meaning of “10% chance of snowing this late afternoon”.
2. For a true or false question, what is the chance of getting the correct answer if
I completely guess the answer? Why?
3. What is the purpose of buying insurance?
4. For a multiple-choice question with four choices, which are A, B, C and D,
what is the chance of getting the correct answer if I completely guess the
answer? Why?
5. Give 2-3 examples of situations where you use probability concept in your
daily life.
6.1 PROBABILITY DISTRIBUTION
Outcome:
Understand the concepts of discrete and continuous random
variables and their associated distributions
Definitions
1. random variable – a variable whose value is determined by the
outcomes of a random experiment
2. discrete random variable – a random variable that can take on any
value from a list of distinct possible values
3. continuous random variable – a random variable that can take on
any value from a continuous range.
4. probability distribution for a discrete random variable – a list of all
the possible values of the random variable and their associated
probabilities
Example1
Which of the following random variables are discrete, and which are
continuous?
a. The number of passengers on a flight from Toronto to Hong Kong.
b. The number of students in Business Statistics class.
c. The time required driving from home to Ryerson University in the morning.
d. The number of traffic fatalities per year in Toronto.
Fall2012 Page # 1 Business Statistics I-QMS102 Chapter5and 6
Probability distributions
1.can be represented by a.table b. graphical chart
2.can be summarized with
a. expected values(or means), which are measures of centre tendency
(location) of the probability distribution
b. standard deviations, which are measures of variability of the probability
distribution
Expected value = mean = E(X) = xP(x)
∑
Standard deviation = ∑ x P(x) − μ2
Example2
The probability distribution for a random variable X is as shown in the following
table.
x 1 2 3 4 5
P(X=x) 0.20 0.25 0.40 0.10 0.05
Determine
a. the expected value b. the variance c. the standard deviation
Example3
The probability distribution for a random variable X is as shown in the following
table.
x 1 2 3 4 5
P(X=x) 0.1 0.3 y 0.2 0.1
Determine
a. the value of y b. the expected value c. the standard deviation
Fall2012 Page # 2 Business Statistics I-QMS102 Chapter5and 6
Example 4
Amanda, the marketing manager for Third Cups Coffee Store is implementing a
new marketing strategy that offers free refills on all coffee order. To review her
strategy, she gathered the following information on coffee refills.
Refills 0 1 2 3 4
Percent 21 37 32 10 0
Compute
a. the probability that a customer refill his/her coffee one or more times
b. the expected value, variance and standard deviation for the distribution of
number of refills.
Example 5
A fast food company plans to install a new ice-cream dispensing unit in one of
the two store locations. The company figures that the probability of a unit being
successful in a location is 5/8 and the annual profit in this case is $185,000. If it is
not successful there will be losses of $36,800. At the location B the probability of
succeeding is ½ but the potential profit and loss are $250,000 and $58,000
respectively.
a. Where should the company locate to maximize expected profit?
b. Which location is less risky, i.e has the lowest relative variability?
Fall2012 Page # 3 Business Statistics I-QMS102 Chapter5and 6
Example6
The normal weekly demand of a certain perishable product sold by Ryerson Inc.
is given by the following distribution
Demand 21 22 23 24 25
Probability 0.4 0.2 0.2 0.1 0.1
The product costs Ryerson $9 each. The product sells for $16 each. If not sold
by the end of the week, the leftover units must be scrapped.
The supplier only has 23, 24 or 25 units for Ryerson to purchase. How many
would you recommend Ryerson to purchase based on expected profit?
23 units
Demand Profit P(X)
21
22
23
24
25
24 units
Demand Profit P(X)
21
22
23
24
25
25 units
Demand Profit P(X)
21
22
23
24
25
Fall2012 Page # 4 Business Statistics I-QMS102 Chapter5and 6
Example 7
You can insure an $80,000 24K gold watch for its total value by paying a
premium of P dollars. If the probability of theft in a given year is estimated to be
0.02,what premium should the insurance company charge if it wants the
expected gain to be $3000?
Fall2012 Page # 5 Business Statistics I-QMS102 Chapter5and 6
Example 8.
Christopher owns an IT business in downtown Toronto. The probability model
below describes the number of employees that may call in sick on any given day.
Number of Employee 0 1 2 3 4 5
P(X=x) a 0.4 0 1.5b 1.5a 0.05
Note: b=2a
What is the expected value of the number of employees calling in sick each day?
Fall2012 Page # 6 Business Statistics I-QMS102 Chapter5and 6
Example 9
A tire manufacturer has four tire designs. The manufacturing cost of each type of
tire is shown below.
Tire Design Fixed Cost/year Variable Cost
A $41,000 $28
B $65,000 $21
C $50,000 $25
D $128,000 $18
There are four possible levels of annual demand: 4,000 tires, 6,000tires, 8,000
tires and 10,000 tires. The respective probabilities are 0.30, 0.25, 0.4, and 0.05.
The selling price will be $73 for A, $71 for B, $ 68 for C and $74 for D.
a. Calculate the expected annual demand for tires
b. Calculate the expected profit for tire Design C
Fall2012 Page # 7 Business Statistics I-QMS102 Chapter5and 6
6.2 THE BINOMIAL PROBABILITY DISTRIBUTION
Outcome:
Recognize situations when the binomial probability distribution
applies, and use formula to calculate binomial probabilities
Binomial probability distribution
1. Often applies when we are interested in the number of times a particular
characteristic turns up. Example: we might want to know how many of the
next 50 customers who come into the store will purchase some
merchandise.
2. The count (how many out of a particular number), which can also be
expressed as a percentage or proportion.
3. This count is a random variable, because its outcome is determined by
chance.
4. A binomial random variable counts the number of times one of only two
possible outcomes takes place (thus the bi-part of the binomial’s name).
5. A binomial experiment must satisfy the following four conditions.
a. there are n identical trials
b. each tria

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