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Probability and Decision Making
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.
The human mind is notoriously bad
at estimating probabilities….
So… let’s start at the beginning!
Sample space: All possible outcomes.
The sum of all outcomes must be
Exhaustive: There are two possible outcomes.
and
Mutually exclusive: The head and tail cannot
come up at the same time.
Three approaches to probability:
1. Classical: Probability is as simple as 1/n
The number of possible outcomes n is not always what we assume.
Suppose a performance review had four ordinal responses:
1) Awful 2) Not good 3) Good 4) Excellent
The average is statistical nonsense, so how can we compare
performance of our employees?
Suppose we simply add 3) and 4) and use the percentage
of all items (or raters) that rate the person positively.
Two workers get exactly the same percent positive,
say 80%.
The number of possible outcomes n is not always what we assume.
Suppose a performance review had four ordinal responses:
1) Awful 2) Not good 3) Good 4) Excellent
The average is statistical nonsense, so how can we compare
performance of our employees?
Suppose we simply add 3) and 4) and use the percentage
of all items (or raters) that rate the person positively.
Two workers get exactly the same percent positive,
say 80%.
Do they have equal performance reviews?
What is the probability that two workers had the same review?
1) Awful
2) Not good
3) Good
4) Excellent
1
2
3
4
20
0
0
80
0
20
0
80
20
0
80
0
0
20
80
0
There are 1,701 different ways that a person
could be rated at 80% !!
Three approaches:
2. Relative Frequency: long –run relative frequency
Three approaches:
3. Subjective: the outcome as we perceive it,
or believe it to be
What is the problem here?
This is the problem with the opponents of evolution.
What is the problem here?
1. If the event has already occurred, therefore the probability of
it occurring is 1.0!
What is the problem here?
1. The event has already occurred, therefore the probability of
it occurring is 1.0!
2. It may be of interest what the probability is of that happening,
IF you are thinking of having to do it again.
Permutations and Combinations
How many ways can n objects
be arranged?
n! = n(n-1)(n-2)… (n-(n-1))
4! = 4 X 3 X 2 X 1 = 24
There are 24 unique ways these people can be arranged as they wait.
Permutations and Combinations
Permutation: How many ways can n objects
be arranged r at a time?
n!
P 
(n  r )!
n
r
How many ways can 4 people be arranged
in unique order in groups of 2?
4!
P 
(4  2)!
n
r
= 24/2 = 12
Permutations and Combinations
Combination: How many unique groups
can be taken from n objects?
n!
C 
r!(n  r )!
n
r
How many ways can 4 people be arranged
in unique groups of 2?
4!
C 
2!(4  2)!
n
r
= 24/(2 X2) = 6
You and your friend Pat, along with two other
people, show up for a job interview.
What is the probability that you will
be selected first for the interview?
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?
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?
What is the probability that you and
Pat will be the first two interviewed?
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?
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?
Blaise Pascal
1623 - 1662
Pascal’s Triangle
Binominal Coefficients
The coefficients are simply the combinations of n objects
taken r at a time…
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.
n
r
r
C pq
nr
120
C .5 .5 
 0.1172
1,024
10
3
3
7
Suppose you write a good resume, so good in fact that 70%
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.
Suppose you write a good resume, so good in fact that 70%
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?
0.7 x 0.7 x 0.7 = 0.343
Suppose you write a good resume, so good in fact that 70%
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 two out of
three assessors must recommend hiring?
3 X 0.147 = 0.441
Or
3!/2! (.7)(.7)(.3) = 0.441
Suppose you write a good resume, so good in fact that 70%
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 anyone
recommends hiring?
1 – 0.33 = 1 – 0.027 = 0.973
Some Interesting Problems
A western US 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?
Some Interesting Problems
Gender Discrimination
1000 men and 1000 women applied to a university,
74% of the men were accepted, but only 26% of
the women are accepted.
Did gender discrimination take place?
No! In fact…. The university discriminated against no one!!
Some Interesting Problems
Gender Discrimination
1000 men and 1000 women applied to a university,
74% of the men were accepted, but only 26% of
the women.
There were two programs:
One excellent: 200 men 800 women applied
One mediocre: 800 men 200 women applied
Some Interesting Problems
Gender Discrimination
1000 men and 1000 women applied to a university,
74% of the men were accepted, but only 26% of
the women.
There were two programs:
One excellent: 10% acceptance rate
200 men 800 women applied
One mediocre: 90% acceptance rate
800 men 200 women applied
Excellent program: 20 men 80 women accepted
Some Interesting Problems
Gender Discrimination
1000 men and 1000 women applied to a university,
74% of the men were accepted, but only 26% of
the women.
There were too programs:
One excellent: 200 men 800 women applied
One mediocre: 800 men 200 women applied
Mediocre program: 720 men 180 women accepted
Some Interesting Problems
Gender Discrimination
1000 men and 1000 women applied to a university,
74% of the men were accepted, but only 26% of
the women.
There were too programs:
One excellent: 200 men 800 women applied
One mediocre: 800 men 200 women applied
In total, 740 men were accepted and only 260 women
Simpson’s Paradox
http://en.wikipedia.org/wiki/Simpson%27s_paradox
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?
Between groups vs Within Groups
When multiple groups make evaluations of the same manager,
the averages will be remarkably consistent.
It is therefore concluded that the evaluation instrument
is a highly reliable measure of the managerial skills of persons
being evaluated.
Is this true?
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?
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?
O. J. Simpson Trial
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?
1. What is the rate of non-wife-beaters who kill their wives?
1. What is the rate of non-wife-beaters who kill their wives?
If the rate is much smaller, then wife-beating is evidence.
1. 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.
1. 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
DNA Evidence.
Suppose a murder was committed in
southern California. DNA is found at
the scene. A computer match finds a
suspect.
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.
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….
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 real probability
that they have the real
murderer based on this evidence?
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?
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!
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.
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 their 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
Elections and Selections
Who wins?
Voting
Method
Tom
Dick
Harry
Winner
Plurality
8
4
12
Harry
Vote-forTwo
18
18
6
Tie Tom
&Dick
Borda
Count
26
22
12
Tom
(Broda Count: first = 2 pts; second = 1 pt, and third = 0 pts)
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.
.
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.
.
Benjamin Disraeli
1804 - 1881
“There are three types of lies:
lies,
damned lies,
and statistics.”
Mark Twain
1885 - 1910