# Probability

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```					Lecture 2
Probability
– some examples
– sources of bias
– types of probability
Problems
Expected Values
Conditional Probability
Important probability formulas
An exercise first
Answer the questions on the sheet of paper
that is handed to you (should take less than
1 minute).
You’re not being tested on your general
knowledge -- I just want to get your
estimate.
Excel functions last week
Calculate:
–   mean using =AVERAGE()
–   median using =MEDIAN()
–   mode using =MODE()
–   standard deviation using =STDEV()
–   variance using =VAR()
–   Get all these at once using >Tools>Data
Analysis> Descriptive statistics
Some thought exercises
about Probability
Birthday Problem

What is the chance of two people in this
class having the same birthday?
Causes of Death

Which is more likely: being killed by
aeroplane parts falling from the sky or being
killed by a shark?
Which is more likely: dying from stomach
cancer or dying in a car accident?
Cancer Screening
A doctor has just examined a woman for breast
cancer. She has a lump in her breast, but based on
years of experience the doctor estimates the odds
of malignancy at 1 in 100. A mammogram is
ordered. In general these tests accurately classify
80% of malignant tumours and 90% of benign
tumours.
If the mammogram says the tumour is malignant,
what would you say the overall chances of the
tumour being malignant are now?
Estimating probabilities:
sources of bias
Representativeness           Conservatism
Availability                 Risk perceptions
Confusion of the Inverse     Anchoring
Looking on the Bright Side   Coincidences
Compound Events

For more details see “The Psychology of Judgement
and Decision Making” by Plous
Report back on homework
How closely did the probability estimates
agree?
Why do you think they differed?
How would you go about coming up with a
single number?
Using pictures to understand
probabilities
Venn Diagrams
– great way to visualise simple probabilities and
overlapping events
Probability Trees
– useful for sequential events, repeated events,
multiple items or people
Contingency Tables
– useful for understanding, visualising and
computing conditional probabilities
Venn Diagrams
Venn diagrams are used to visually convey the
concept of sample spaces and events.
A rectangle is used to represent the sample
space of an experiment and as such contains all
possible sample points.
Circles within the rectangle represent events.
The area within the circle contains the sample
points which belong to that event.
Simple Venn Diagram

A        Ac
Venn Diagram

A             B

A  Bc     A B    Ac  B

Ac  Bc
Venn Diagram of Union of
Events A and B

AB

Ac  Bc
Share price problem
P(A up) = 0.7,     P(B up) = 0.5
P(C up) = 0.3, P(A up  B up) = 0.3
P(A up  C up) = 0.1,
P(B up  C up) = 0.2
P(A up  B up  C up) = 0.05
What is P(A up  B up) ?
What is P(A up  B up  C up) ?
What is P(A up  B down  C down) ?
Probability Trees
Each node is an experiment. Each branch is
an outcome. Branches that come out of a
node should be exhaustive and mutually
Conditional
exclusive.                          probabilities
Prob= 0.7
Prob= 0.6                      YY 0.42
Y
0.6   Prob= 0.3
YN 0.18
1                         Prob= 0.4
Prob= 0.4                      NY 0.16
N
0.4   Prob= 0.6
NN 0.24
Job offers
Joint Probability Table

Event
A                 Ac

B    P(A  B)            P(Ac  B)    P(B)
Event
Bc   P(A  Bc)           P(Ac  Bc)   P(Bc)

P(A)               P(Ac)       1
Convert Cross-Tabs Table Into
a Joint Probability Table
No JS    JS

Cross-Tabs    LOW
13      4
PROD
Table
(n = 36)     HIGH
PROD    1      18
Total = 36

No JS    JS

Joint         LOW
.361    .111     4/36
PROD
Probability
Table         HIGH
PROD   .028    .500
Joint and Union Probabilities
No JS     JS
LOW
PROD     .361    .111    .472
HIGH
PROD     .028    .500    .528
.389      .611 1.00
Unconditional Probability
P(LOW PROD)          = .472
Joint Probability
P(LOW PROD  JS) = .111
Union Probability
P(LOW PROD  JS) = .361 + .111 + .500 = .972
Excel - Probability

Pivot-tables
Downloadable spreadsheet
from webpage with
“proforma” probability tables
Scholarships
In a study of 100 students who had been
awarded university scholarships, it was
found that 40 had part-time jobs, 25 had
achieved honours in the previous semester,
and 15 had both a part-time job and had
achieved honours.
What was the probability that a student had
a part-time job or had achieved honours (ie.
the union probability)?
Expected Value
What is an expected value?
Notation: expected value of x is E(x)
Calculating E(x):

E ( x)  P( x1 ) x1  P( x2 ) x2  P( x3 ) x3  ...
Expected Value example
If you want to rent out a house and you know:
Weekly Rent          Probability
\$210                 0.4
\$225                 0.3
\$240                 0.25
\$250                 0.05

What do you expect to earn in rent each week?
Watch this space (Video)
Use probability language to describe the
two main positions in this argument
Which side do agree with? Why?
Can you think of other examples where
there are similar differences in interpreting
probabilities and expected values?
Monty Hall/Let’s Make a Deal

Door A    Door B     Door C
Conditional Probability
P(A|B) is the probability of A happening
given that B has happened.
This is called a conditional probability
Important formula: (Bayes’ Theorem)
P(A|B) = P(A  B) / P(B)
Or round the other way:
P(A  B) = P(A|B) x P(B)
Unconditional and Conditional
Probabilities
No JS      JS
LOW
.361     .111    .472
PROD
HIGH
PROD
.028    .500     .528

.389     .611
Unconditional Probability
P(HIGH PROD)           = .528
Conditional Probability
P(HIGH PROD | JS) = .500/.611 = .81
Only Interested in JS Groups. Thus .611 is the
Denominator. High Prod is .500, the Numerator.
Independent Events

Events are independent if the outcome of one
does not affect the outcome of the other.
Example:
– the profit made by Telstra is independent of what
I chose to have for breakfast yesterday.
– But the profit made by Telstra is not independent
of whether I chose to phone my friend in
California yesterday.
Statistical Independence ?
If P(A) = P(A | B), then Events A and B
are Independent.
– Does P(High Prod) = P(High Prod | Job
Switch)?

Knowing Event B Happened Does Not
Affect the Probability of A.
Is Job Switching Related to
Productivity Level?
P(HIGH PROD)       = .528
P(HIGH PROD | JS) = .500/.611 = .818
P(HIGH PROD | NJS) = .028/.389 = .072
Note: Three Probabilities are Different.
 Probability of High Productivity if Group Members
Switch Jobs is .818.
 Probability of High Productivity if Group Members
Do Not Switch Jobs is Only .072.
 Conclusion: High Productivity and Job Switching May
be Related.
Probability Trees – independent
events
Trees can also be used when events are
independent.
Probabilities on branches are just simple
probabilities of each event occurring
This is because literal (symbolic) definition
of independence is: P(A) = P(A | B)
Conditional probability is simple
probability for independent events
Mutually Exclusive Events

Events are mutually exclusive if they both
couldn’t happen at once.
Example:
– Telstra making a profit of more than \$300 million is
mutually exclusive with them making a profit of less
than \$300 million this year.
– Me having toast for breakfast is not mutually
exclusive with me having Weetbix for breakfast (I
might have both).
Calculating Union probabilities
In general the probability of A  B
happening is:
P(A  B) = P(A) + P(B) - P(A  B)

If events are mutually exclusive then the
probability of A  B happening is:
P(A  B) = P(A) + P(B)
Calculating Intersection
probabilities
In general the probability of A  B
happening must be found by gathering data
to estimate it separately.

However if events are independent then the
probability of A and B happening is:
P(A  B) = P(A)  P(B)
Murder He Wrote
Do you understand the probabilities that are
talked about in the article?

What does the law mean when it talks about
“beyond a reasonable doubt”? Can you
assign a probability to this?
The Birthday Problem
Birthday Problem

1
0.9
0.8
P(at least one match)

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
13

16
19

22
25
28

31
34

37
40
43

46
49

52
55
58
1
4

7

Number of People (n)
Birthday Problem (the
calculations)
Probability of 2 people not having the same
birthday = 364/365
Probability of 3 people not having the same
birthday = Probability of 2 people not
having the same birthday AND a 3rd person
not having the same birthday as either of the
first 2 = 364/365  363/365
Probability of 4 people not having the same
birthday = 364/365  363/365  362/365
Birthday problem (cont.)
Prob of no birthday matches with (n+1)
people =        364  363 362  ... (365  n)
365n
Prob of (n+1) people having at least one
matching birthday = 1 - P(no matches)
Example: P(no matches for 10 people) = 0.8831
P(at least one match for 10 people) = 0.1169
Wallet Problem

What is the
5    5        probability that given
we select a five dollar
bill from one of these
5   10       three wallets at
random that the other
bill in that wallet is a
10   10       five?
Probability Trees
Remember we used probability trees with
conditional probabilities on the branches.
Useful to depict sequential events.
P(C|A)=0.4
AC 0.08
P(A)=0.2    A
0.2   P(D|A)=0.6
AD 0.12
1                     P(C|B)=0.3
BC 0.24
P(B)=0.8    B
0.8   P(D|B)=0.7
BD 0.56
Conditional probabilities
Can also think of a conditional probability as a
probability of something happening within a
“subset” of the population.
Use a Venn diagram or a probability table e.g.

P(A | B) = 10 / 50 = 0.2
A        B
20   10   40   (Notice that this is identical to
30
Bayes theorem formula)
Another way of thinking about
conditional probabilities
Many times during probability analysis, we
may want to revise some prior probabilities
P(A) based on new information (B).
Bayes Theorem allows us to do this. The
new probabilities are called posterior
probabilities P(A|B) and, as the new
estimates of the likelihood of A, can be used
in future calculations instead of P(A).
Excel – Bayes Theorem

Download Bayes Theorem
spreadsheet from webpage to
do the calculations
Important Formulas

P ( A)  P ( A )  1
c

P ( A  B )  P ( A  B )  P ( A)
c

P ( A  B )  P ( A)  P ( B )  P ( A  B )
P ( A B )  P ( B  A)       P( B)
P( A  B)  P( B) P( A B)
Rashes
A pharmaceutical company conducted a study to
evaluate the effect of an allergy relief medicine; 250
patients with symptoms that included itchy eyes and
a skin rash were given a new drug. 90 of the patients
experienced eye relief, 135 had their rash clear up,
and 45 had relief from both conditions.
What is the likelihood that a patient will experience
relief from at least one of these conditions?
Do these events appear independent? Explain.
Rush Orders
Rush orders for a raw material are placed with two
different suppliers, A and B. If neither order arrives
in 4 days production is shut down until at least one
comes. P(A will deliver in 4 days) = 0.55,
P(B will deliver in 4 days) = 0.35
P(both deliver in 4 days)?
P(at least one delivers in 4 days)?
P(production is shut down)?
Advertising Soap
B = bought, S = remembers seeing ad
P(B) = 0.2, P(S) = 0.4, P(B  S) = 0.12
P(B|S)? Does seeing the ad increase probability of
purchasing? Would you keep advertising?
If those who don’t buy from you, buy from your
competitor:
What would be your estimate of your market
share?
Would your continuing the ad increase your
market share? Why or why not?
Advertising Soap (cont.)
Another ad has been tested and has values of
P(S)=0.3 and P(B  S) = 0.1.
What is P(B|S) for this ad?
Which ad has a bigger effect on purchases?
Credit?
A bank is reviewing its credit card policy with a view
to recalling some of its credit cards. In the past
approximately 5% of cardholders have defaulted and
the bank has been unable to collect the outstanding
balance. The bank has further found that the
probability of missing one or more payments for those
customers who do not default is 0.20. Of course the
probability of missing one or more payments for those
who default is 1.
Given that a customer has missed a payment, compute
the probability that they will default.
Integrated Circuit chips
A process produces IC chips. Over the long-run
the fraction of bad chips produced by the process
is around 20%. Thorough testing is expensive but
there is a cheap test. All good chips pass the cheap
test but so do 10% of bad chips.
Given that a chip passes the test what is the
probability that it is a good chip?
If a company sold all chips that passed the test
what fraction of what they sold would be bad?
Cancer Screening
Use Bayes Theorem to answer the cancer
screening question from beginning of lecture.
Prior probability of malignancy = 0.01
P(positive test | malignant) = 0.8
P(negative test | benign) = 0.9
What is the posterior probability of malignancy?
(I.e. the new probability of malignancy if you get
information that the test was positive)
What did we do?
Discussed where intuition about probability fails
Discussed how probability is estimated
Venn Diagrams
Explored the use of tree diagrams and probability
tables in calculating probability.
Talked about independence and mutual exclusivity
Expected value
Conditional probability
Talked about statistical independence
Worked on problems
Excel functions today
Expected values using =SUMPRODUCT()
Bayes theorem and probability tables
spreadsheet – download and use to
automate simple calculations. Practice using
this to solve problems you get in class and
in textbook/webpage.
Managerial applications
What did you learn today that makes a
difference to the way you manage?
What are the three most important things to
remember from today’s lecture?
Next class
Prepare case study “Who is the customer?”
(Specific questions given on the class
webpage). This is assessed if your syndicate
chose option 1
Read and answer questions for “Hair Tonic”
Read supplementary readings on Decision
Trees and Valuing Information
Answer additional questions on webpage

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