OPRE504
Chapter Study Guide
Chapter 7
Randomness and Probability
Terminology of Probability
For a Random phenomenon, there are a number of possible Outcomes. For example, tossing a
coin could result in either a head or a tail (a total of two possible outcomes). Tossing is called a
Trial. A trial generates an outcome. An Event is a collection of possible outcomes. Sample
Space is the special Event which contains all possible outcomes.
Theoretical (Model-based) Probability of a Random Phenomenon:
# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴
P (A) = 𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Probability Rules:
Rule 1
0 ≤ P (A) ≤ 1
In a two-outcome situation, 50% means that two outcomes are equally likely
Rule 2
If a random phenomenon has N possible outcomes, then
P (outcome 1) + P (outcome 2) + … P(outcome N) = 1
Rule 3
Complement Rule
P (A) = 1 – P (Ac) [ P (Ac) is the probability of Event A is not occurring.]
Rule 4
Multiplication Rule for Independent Events A and B
Probability of all events occurring simultaneously is the product of the
probabilities of all individual events
P (A and B) = P (A) x P (B)
Rule 5
Addition Rule for Disjoint (Mutually Exclusive) Events A and B
Probability of either of the two disjoint events occurring
P (A or B) = P (A) + P (B)
Rule 6
General Addition Rule for Any Two Events A and B
Probability of either of the two events occurring is the sum of the probabilities of
two individual events subtracted by the potential double counting of both events
happening simultaneously.
P (A or B) = P (A) + P (B) – P (A and B)
If A and B are mutually exclusive, P (A and B) = 0, Rule 6 connects Rule 5:
so P (A or B) = P (A) + P (B) + 0 = P (A) + P (B)
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Marginal Probability and Joint Probability in A Contingency Table
An Example of Contingency Table
Note: A customer is randomly
selected to receive a prize by
an electronic retailer
Customer
TABLET COMPUTER PREFERENCE
Total
iPad
Android
Other
Male
120
100
30
250
Female
80
40
30
150
200
140
60
400
Total
The probability of selecting a male customer is called _______________:
P (Male) = __________ because two numbers (250 and 400) in the margins of the contingency
table are used.
The probability of selecting a customer who is male and prefers iPad is called ____________:
P (Male and iPad ) = _______________
The probability for a male customer to choose iPad is called _______________:
P (iPad | Male) = _______________________________________ because it describes the probability of
preferring an iPad given a male customer.
Additional Probability Rules
Rule 7
General Multiplication Rule for Any Two Events: A and B
P (A and B) = P (A) x P (B|A) or P (A and B) = P (B) x P (A|B)
Rule 8
Independence Rule
If A and B are independent, we would expect that
P (A) = P (A|B), probability of A does not change whether B occurs or not.
P (B) = P (B|A), probability of B does not change whether A occurs or not.
To evaluate whether Events A and B are independent, we can also check whether
P (A and B) = P (A) x P (B)
Question 7.1 [ Sharpe 2011, Exercise 11, p.198] In developing their warranty policy, an
automobile company estimates that over a 1-year period 17% of their new cars will need to be
repaired once, 7% will need repairs, and 4% will require three or more repairs. If you buy a new
car from them, what is the probability that your car will need:
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a)
Some repair?
b)
No repairs?
c)
No more than one repair?
Question 7.2 [Sharpe 2011, Exercise 33, p.200] A GfK Roper Worldwide survey in 2005 asked
consumers in five countries whether they agreed with the statement “I am worried about the
safety of the food I eat.” Here are the responses classified by the age of the response:
Agree
13-19
20-29
Age Group 30-39
40-49
50+
Total
661
816
871
914
966
4228
Neither
Agree nor
Disagree
368
365
355
335
339
1762
Disagree
452
336
290
266
283
1627
Don’t
Know / No
Response
32
16
9
6
10
73
Total
1513
1533
1525
1521
1598
7690
If we select a person at random from this sample:
a)
What is the probability that the person agreed with the statement?
b)
What is the probability that the person is younger than 50 years old?
c)
What is the probability that the person is younger than 50 and agrees with the statement?
d)
What is the probability that the person is younger than 50 or agrees with the statement?
e)
What is the probability that the person agrees is aged between 20 and 29?
f)
Are response and age independent?
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You can choose any cell to test whether response and age are independent. If they are
independent, for example, we expect that P(agree) = P (agree | 13-19) and P (agree and 13-19) =
P (agree) x P (13-19).
1).
Check if P(agree) = P (agree | 13-19):
2) Alternatively, check if P (13-19 and agree) = P (13-19) x P (agree)
Question 7.3 [Sharpe 2011, Exercise 43, p.202] In a real estate research, 64% of homes for sale
have garages, 21% of homes have swimming pools and 17% have both features.
a) What is the probability that a home for sale has a garage, but not a pool?
b) If a home for sale has a garage, what’s the probability that it has a pool, too?
c) Are having a garage and a pool independent events? Explain.
Check whether P(Pool | Garage ) = P (Pool)?
Or check whether P (Pool and Garage) = P (Pool) x P (Garage)?
d) Are having a garage and a pool mutually exclusive? Explain.
More exercises:
Sharpe 2011, Guided Example, M&M’s Modern Market Research, pp.183-185
Sharpe 2011, Chapter 7, Exercises 9, 10, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29,
30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50.
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Probability Tree
Question 7.4 [Sharpe 2011, Exercise 54] Extended warranties.
A company that manufactures and sells consumer video cameras sells two versions of their
popular hard disk camera, a basic camera for $750 and a deluxe version for $1250. About 75%
of customers select the basic camera. Of those, 60% purchase the extended warranty for an
additional $200. Of the people who buy the deluxe version, 90% purchase the extended warranty.
a)
Sketch the probability tree for total purchases.
b)
What is the percentage of customers who buy an extended warranty?
c)
What is the expected revenue of the company from a camera purchase (includes warranty
if applicable)?
d)
Given that a customer purchases an extended warranty, what is the probability that he or
she bought the deluxe version?
More Exercises:
Sharpe 2011, Chapter 7, Figures 7.2, 7.3 and 7.4, pp.190-192
Sharpe 2011, Chapter 7, Exercises 51, 52, 53, 55, and 56
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