Probability and Decision Making

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Probability and Decision Making
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Many problems with decision making can be made
less stressful with an understanding
of probability.
But , almost nothing can be more confusing,
and create more errors than an incorrect
application of probability.
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The human mind is notoriously bad at estimating
probabilities.
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In a survey of faculty at the University of
Nebraska, 68 rated themselves in the top 25 for
teaching ability.
In a similar survey, 87 of MBA students at
Stanford University rated their academic
performance as above the median.
For driving skill, 93 of the US sample put
themselves in the top 50 (above the median).
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Chance of winning the lottery with 6 correct
numbers (1-49) is 1 in 13,983,816
Chance of dying in a car accident is 1.49 in 100
A person is 21,000 times more likely to die in a
car than win the lottery!
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Sample Space All possible outcomes The sum of
all outcomes must be Exhaustive and Mutually
exclusive.
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Three approaches to probability
Classical Probability is as simple as 1/n
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Three approaches Classical Probability is as
simple as 1/n Relative Frequency long run
relative frequency
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Three approaches Classical Probability is as
simple as 1/n Relative Frequency long run
relative frequency Subjective the outcome as
we perceive it, or believe it to be
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Permutations and Combinations How many ways can
n objects be arranged?
n! n(n-1)(n-2) etc 4! 4 X 3 X 2 X 1 24
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There are 24 unique ways these people can be
arranged as they wait.
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Permutations and Combinations Permutation How
many ways can n objects be arranged r at a time?
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How many ways can 4 people be arranged in unique
order in groups of 2?
24/2 12
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Permutations and Combinations Combination How
many unique groups can be taken from n objects?
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How many ways can 4 people be arranged in unique
order in groups of 2?
24/(2 X2) 6
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You and your friend Pat, along with two other
people, show up for a job interview.
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What is the probability that you will be selected
first for the interview?
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What is the probability that you will be selected
first for the interview? What is the probability
that the four of you will line up at random, and
you are at the head of the line?
Each of you are given a number (1 4), what is
the probability that the four of you will line up
at random, and the number given to each will be
in the order of 1, 2, 3, 4?
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What is the probability that you and Pat will
be the first two interviewed?
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What is the probability that you and Pat will
be the first two interviewed? What is the
probability that you and Pat will be interviewed
first, and you will be interviewed before Pat?
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Blaise Pascal 1623 - 1662
Pascals Triangle Binominal Coefficients
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The coefficients are simply the combinations of
n objects taken r at a time
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Suppose you wanted to know the probability of
getting exactly 3 heads in ten tosses of a fair
coin? The probability of a head is p ½, the
probability of a tail is q ½.
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Suppose you write a good resume, so good in fact
that 80 of the recruiters who read it would
offer you a job. Suppose further that you apply
to a company that utilizes three independent
resume assessors.
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Suppose you write a good resume, so good in fact
that 80 of the recruiters who read it would
offer you a job. Suppose further that you apply
to a company that utilizes three independent
resume assessors. What is the probability of
getting this job if all three assessors must
recommend hiring?
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Suppose you write a good resume, so good in fact
that 80 of the recruiters who read it would
offer you a job. Suppose further that you apply
to a company that utilizes three independent
resume assessors. What is the probability of
getting this job if at least two out of three
assessors must recommend hiring?
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Some Interesting Problems A western city is 30
Hispanic, 45 of all convictions for spousal
abuse are Hispanic. An activists claims that
this is evidence that jurists are biased against
Hispanics. Is this true?
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Some Interesting Problems Gender
Discrimination 1000 men and 1000 women applied to
a university, 70 of the men were accepted, but
only 40 of the women. Did gender discrimination
take place?
Yes! In fact. The university discriminated
against men!!
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Some Interesting Problems Gender
Discrimination 1000 men and 1000 women applied to
a university, 70 of the men were accepted, but
only 40 of the women. There were two
programs One excellent 200 men 800 women
applied One mediocre 800 men 200 women applied
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Some Interesting Problems Gender
Discrimination 1000 men and 1000 women applied to
a university, 70 of the men were accepted, but
only 40 of the women. There were two
programs One excellent 200 men 800 women
applied One mediocre 800 men 200 women
applied Excellent program 20 men 25 women
accepted
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Some Interesting Problems Gender
Discrimination 1000 men and 1000 women applied to
a university, 70 of the men were accepted, but
only 40 of the women. There were two
programs One excellent 200 men 800 women
applied One mediocre 800 men 200 women
applied Mediocre program 82.5 men 100
women accepted
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Some Interesting Problems Gender
Discrimination 1000 men and 100 women applied to
a university, 70 of the men were accepted, but
only 40 of the women. There were too
programs One excellent 200 men 800 women
applied One mediocre 800 men 200 women
applied In total, 700 men were accepted and
only 200 women
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Simpsons Paradox
http//en.wikipedia.org/wiki/Simpson27s_paradox
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Leniency vs Reciprocity
Are employees who work for companies with higher
pay more likely to be happy with their jobs?
A study of 500 companies found no relationship
between average employee pay and average workers
happiness scores. The report concludes that
higher paid workers are NOT more happy with
their jobs.
Is this true?
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Some Interesting Problems A witness saw a
hit-and-run. He claims the car was a city cab
painted yellow. There was only one yellow cab in
service at the time. Under the same conditions
the witness was found to be able to see the
difference between yellow and white cabs 80 of
the time. The prosecutor tells the jury that
they must convict the cab driver because the
evidence is overwhelming. Is it?
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It turns out that there is 20 yellow cabs in the
city and 80 white cabs. What is the probability
that the witness correctly identified the color
of the car?
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O. J. Simpson Trial
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Johnnie Cochran argued that evidence that O.J.
beat his wife was irrelevant because only 1 in
1,000 wife-beaters went on to kill their wives.
What is the problem?
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• What is the rate of non-wife-beaters who kill
  their wives?

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• What is the rate of non-wife-beaters who kill
  their wives?
• If the rate is much smaller, then wife-beating
  is evidence.

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• What is the rate of non-wife-beaters who kill
  their wives?
• If the rate is much smaller, then wife-beating
  is evidence.
• 2. O.J.s wife was actually murdered.

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• What is the rate of non-wife-beaters who kill
  their wives?
• If the rate is much smaller, then wife-beating
  is evidence.
• 2. O.J.s wife was actually murdered.
• The real question is
• If a wife is murdered, what is the probability
  that she had
• previously been beaten?

From Dr. Michael Starbird University of Texas
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DNA Evidence.
Suppose a murder was committed in southern
California. DNA is found at the scene. A
computer match finds a suspect.
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DNA Evidence.
Suppose a murder was committed in southern
California. DNA is found at the scene. A
computer match finds a suspect. The prosecuting
attorney tells the jury that there is only one
chance in a million that two people would have
the same DNA evidence.
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DNA Evidence.
Suppose a murder was committed in southern
California. DNA is found at the scene. A
computer match finds a suspect. Therefore they
must convict.
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DNA Evidence.
Suppose a murder was committed in southern
California. DNA is found at the scene. A
computer match finds a suspect. What is the
actual probability that they have the
real murderer based on this evidence?
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An airport screening devise is so accurate that
it will make an error only one time in
500,000! What will be the probability that a
person accused of having a weapon is falsely
accused?
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There are about 800 million passenger flights in
the United Sates per year.
Fifty of these were terrorists trying to get a
weapon on an airplane.
Assuming that the screening was 100 successful
at finding the terrorists, the probability of
being falsely accused is (1,600-50)/1,600 or
0.969!
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Elections and Selections
An important committee in state government has
22 members. They must pick a chair. This person
will will have immense power over how the state
will spend its budget.
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Elections and Selections
An important committee in state government has
22 members. They must pick a chair. This person
will will have immense power over how the state
will spend its budget. Here is there selection
Rank Big Eight Smart 4 Wise 6 Prac 6
First Tom Dick Harry Harry
Second Dick Tom Dick Tom
Third Harry Harry Tom Dick
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Elections and Selections
Who wins?
Voting Method Tom Dick Harry Winner
Plurality 8 4 12 Harry
Vote-for-Two 18 18 6 Tie Tom Dick
Borda Count 26 22 12 Tom
(Broda Count first 2 pts second 1 pt, and
third 0 pts)
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Elections and Selections
Pair-Wise Sequential Voting The Bubble-Up The
candidates are put into some sequential order,
then an election between 1 and 2, then winner and
3, and then the winner against 4. Etc. .
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Elections and Selections
Pair-Wise Sequential Voting The Bubble-Up The
candidates are put into some sequential order,
then an election between 1 and 2, then winner and
3, and then the winner against 4. Etc. This
method can select a winner that no single voter
preferred. .
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Benjamin Disraeli 1804 - 1881
There are three types of lies lies,
damned lies, and statistics.
Mark Twain 1835 - 1910