School

Miami University **We aren't endorsed by this school

Course

ISA 225

Subject

Statistics

Date

Aug 30, 2023

Pages

4

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Question 1:
Rolling a three-sided die twice.
a)
Identify the sample space
S = { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) }
b)
Find the probability of a sum of 4.
S = { (1,1), (1,2),
(1,3),
(2,1),
(2,2)
, (2,3),
(3,1)
,
(3,2), (3,3) }
P(sum=4) = 3/9= 1/3
c)
If we consider that a sum greater than 5 results in a win. Then identify the sample space of winning, and find the
winning probability.
S(sum>5) = {(3,3)}
P(winning) =1/9
Question 2:
Determine which of the following number(s) could represent the probability of an event. (Multiple Answers)
a)
0
b) 1.5
c) -1
d) 50%
e) 2/3
Question 3-4: Ture or False
3. __
False
___
After five tosses of a coin repeatedly producing heads, I believe that this coin is not a fair coin, which
means that the probability of head and tail is not half-half.
4. __
False
__
An investment newsletter makes general predictions about the economy to help their clients make sound
investment decisions. They advised buying a stock that had gone down in the past four sessions because they said that it
was clearly "due to bounce back."
Question 5:
Identify the probability type for each of the following description:
Theoretical
P(D=Friday) = 1/5 = 0.2 if D is a randomly select weekday.
Theoretical
P(Tail & Head) = P(T) * P(H) = 0.5 * 0.5 = 0.25 if we flip a fair coin twice.
Empirical
Meteorologists running simulations to predict the probability that a severe storm will make landfall.
Empirical
Collecting information after implementing an ad campaign and using the results to estimate the
probability that a consumer will buy a product.
Subjective
You look at the sky and assume the probability of it raining.
Subjective
What you think the chance of becoming sick would be if you eat some left-overs in the refrigerator.

Question 6:
Consider rolling a fair six-sided die once with the two possible events,
Event A:
roll an even number
Event B:
roll a number less than or equal to 3
State the Sample Space of the whole experiment:
S = { 1, 2, 3, 4, 5, 6}
State the Sample Space of the Event A:
S(A) = {2, 4, 6}
State the Sample Space of the Event B:
S(B) = {1, 2, 3}
Draw a Venn diagram in the scratch paper, and find the following probability:
P(A)= 3/6 = 0.5
P(B)= 3/6 = 0.5
P (A and B) = 1/6
P (A or B) = 5/6
Question 7: Constructing Contingency Tables
Sometimes we're given probabilities without a contingency table. Although a table isn't always necessary to perform
calculations, it is highly suggested that you create one to help keep the information organized.
Example: Facebook reports that 70% of their users are from outside the United States and that 50% of their users log on
to Facebook every day. Suppose that 20% of their users are United State users who log on every day.
a)
What percentage of Facebook's users are from the United States?
1-70% = 30%
b)
What type of probability is the 20% mentioned above?
Joint Probability
c)
Construct a contingency table showing all the joint and marginal probabilities.
U.S.
Outside U.S.
Yes
0.2
0.3
0.5
No
0.1
0.4
0.5
0.3
0.7
1
d)
What is the probability that a user is from the United States given that he or she logs on every day?
U.S.
Outside U.S.
Yes
0.2
0.3
0.5
No
0.1
0.4
0.5
0.3
0.7
1
P(U.S. | logon everyday) = 0.2/0.5 = 0.4
e)
Are being from the United States and logging-on every day independent? Explain.
P(U.S) = 0.3
P(U.S. | logon everyday) = 0.4,
They are not equal,
then logon everyday can affect the U.S.
Then being from the United States and logging-on every
day are NOT independent.

Question 8: Drug Testing of Air Traffic Controllers
ATCs are required to undergo periodic random drug testing. A simple, low-cost urine test is used for initial screening. It
has been reported that this particular test has a sensitivity and specificity of 0.96 and 0.93. This means that if there is
drug use, the test will detect it 96% of the time. If there is no drug use, the test will be negative 93% of the time. Based
on historical results, the FAA reports that the probability of drug use at a given time is approximately 0.007 (this is called
the prevalence of drug use).
Draw a probability tree for the situation. (Setup as: Drug Use -> Test Result -> Joint Probabilities), answer the following
questions:
a) A positive test result puts the air traffic controller's job in jeopardy. What is the probability of a positive test
result?
P(+) = P(+ & drugs) + P(+ & no drugs) = 0.00672 + 0.06951 = 0.07623
b) Find the probability an air traffic control truly used drugs, given that the test is positive.
P(used drugs |+) = 0.00672/ 0.07623 = 0.08815
Question 9:
Suppose a supermarket database has 100,000 point-of-sale transactions. Of the transactions, 2000 include both orange
juice and OTC flu medication, and 800 of these include soup.
a)
Find the Support and Confidence for the association rule
(OJ and Flu med)
(soup)
Support =
# ୟ୬ୢ ୪୳ & ୗ୭୳୮
ୟ୪୪ ୲୰ୟ୬ୱୟୡ୲୧୭୬ୱ
=
଼
ଵ,
= 0.008
Confidence =
# ୟ୬ୢ ୪୳ & ୗ୭୳୮
# ୟ୬ୢ ୪୳
=
଼
ଶ
= 0.4
b)
If 1000 of all transactions had soup, find the Lift Ratio for the association rule
(OJ and Flu med)
(soup)
Lift = Confidence/ Benchmark confidence
Where confidence =0.4, and Benchmark confidence = #Soup/ all transaction = 1000/100,000= 0.01
Then Lift =0.4/0.01 = 40

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