1.1 Problem
What we know:
●​ P(Positive Test | Test) = 0.95
●​ P(Negative Test | NonUser) = 0.90
●​ P(User) = 0.15
●​ P(NonUser) = 0.85
●​ 0.05 = False Negative
●​ 0.10 = False Positive
1. What is the probability of a positive test?
P(Positive Test)=(0.95⋅0.15)+(0.10⋅0.85)=0.1425+0.085=0.2275 = 22.75%
Edit: What are the possibilities of getting a positive result.
1.​ 95 of the 10%
2.​ And 85 of the 10%
2. You test someone and the test is positive. What is the probability that the tested
individual uses this illegal drug?
P(User∣Positive Test) = P(Positive Test∣User) X P(User)​/ P(Positive Test)
0.95 X 0.15 / .2275 = 0.626 = 62.64%
.95 of the 15 / the true positive number from above .2275
3. Construct a table of the joint probabilities with test results on the column margin and
user/non-user on the row margin.
Status
Positive Test
Negative Test
Total
User
.1425
.0075
0.15
Non User
.0850
.7650
0.85
Total
.2275
.7725
1.00
4. What is the probability of obtaining a false negative for this test (e.g. the probability
that the individual tested is a user given a negative test)?
P(User∣Negative Test)=.05X0.15 / 0.77250 ​= 0075 / 0.77250. ​≈0.0097 = .97%
5. What is the more informative result: positive test or negative test? Positive
Negative
1.2 Problem
You are asked to evaluate a program that comes from the head of Global
Sales. Entertainment budgets are argued frivolous expenditures that only
set up employees for enjoyment and do not impact sales. We have a total
of 100 sales people that have been chosen because they represent identical
account profiles and they were also randomly assigned to the group with
an entertainment budget or without one such that P(Entertainment Budget=
Yes) = P(Entertainment Budget=No) = 0.5. The outcome of interest is
a sale (Yes or No).
Sales
Entertainment
Budget
Y
N
total
Y
40
10
50
N
30
20
50
total
70
30
100
1.​ Complete the above contingency table.
2.​ Are entertaining and selling independent? No
3.​ What is the probability that a sale occurs given that an individual has an
entertainment budget? 80%
a.​ 40/100=0.4
b.​ 50/100=0.5
c.​ 0.4/0.5=0.8
i.​ Easier method 40/50=0.8
4.​ What proportion of all sales come from those without entertainment budgets?
a.​ 30/100=0.3
b.​ 70/100=0.7
c.​ 0.3/0.7=0.429 = 42.9%
i.​ Correct answer
ii.​ 30 out of 70 sales come from those without ent. budget.
5.​ Can you quantify when entertainment budgets are a worthwhile investment?
a.​ Yes
6.​ What quantity or quantities would we need to know to evaluate their usefulness?
a.​ Sales given no entertainment budget.
b.​ 30/100=0.3
c.​ 50/100=0.5
d.​ 0.3/0.5=0.6 = 60%
1.3 Problem 3
One of the most important metrics for Facebook is something known as
a Markox matrix – a matrix linking current and past realizations of some
outcome. Their particular interest is in the use of an experimental Facebook
shopping platform per month. Consider the following subset of data
for a recent month. There are a few features of this platform that make it
unlike most others; it is by invitation only so the size of both the yes and
no populations are actually known (this is almost never true in any precise
way).
​
​
​
​
​
​
Current User
Prior User
Y
N
total
Y
164000
20000
184000
N
24000
1952000
1976000
Total
188000
1972000
2160000
1.​ What is the conditional probability of being a current user of the shopping
platform given that you previously used the platform?
a.​ 164000/184000=0.891=89.1%
2.​ What is the rate of “user loss” – the conditional probability of becoming a
non-user of the platform given use in the prior period?
a.​ 20000/184000=0.11 = 11%
3.​ Is the platform growing or shrinking? How would you measure that?
a.​ 188000-184000= 4000/184000= 0.0217 =2.17%
b.​ Yes, it’s growing.
i.​ Other method
ii.​ 188000-184000=4000. Grew by this amount.
4.​ Explain/defend your answer?
a.​ We want to find new users to see the growth of the platform. So we take
current users and current users that were prior users, but we don’t want to
double count so we subtract.
5.​ What is the probability of not being a user of the platform in the current period?
a.​ 1972000/2160000=0.913 = 91.3%
1.4 Problem 4
A machine produces defective parts with probabilities that depend on the
state of repair. If the machine is in good working order, it produces defective
parts with probability 0.02. If the machine is wearing down, it produces
defective parts 10% of the time. If the machine is in need of repairs, then it produces
defective parts 30% of the time. The probability that the
machine is in good working order is 0.8; the probability that the machine
is wearing down or in need of repair is 0.1 (for each).
What we know:
●​ P(defective parts) = .02
●​ P(defective parts | wearing down) = 0.10
●​ P(defective parts | need repair) = 0.30
●​ P(works) = 0.8
●​ P(wearing down) = 0.1
●​ P(need repair)= 0.1
1. Compute the probability that a randomly selected completed part is
defective. NB: There is no difference in the rate of production among
the repair states, only a difference in the number of defects.
P(defective)= (0.02x0.8) + (0.10x0.10) + (0.30x0.10)
P(defective) = 0.016 + 0.01 + 0.03 = 0.056= 5.6%
2. A factory consists of three machines. One machine is in good working
order. One machine is wearing down and one is in need of repairs.
You are given a defective part; what is the probability that each machine
produced it?
P(works | defective) = 0.02x0.8= 0.016/0.056= 0.2857 =28.57%
P (wearing down | defective)= 0.10x0.10=0.01/0.056= 0.1786 = 17.86%
P(need repair | defective)= 0.30x0.10=0.03/0.056= 0.5357 = 53.57%
Correct
P(defective)= (0.02x 0.333) + (0.10x 0.333) + (0.30x 0.333)= 0.1398
Defective
Good
.02(D|G)
.8 P(G)
Wearing
Need Repair
0.16
.01
.03
.784
.09
.07
.8
.1
.1
.98
.8
NonDefective