Probability and Decision Making

Many problems with decision making can bemade less stressful with an understanding of

probability.

But , almost nothing can be more confusing, andcreate more errors than an incorrect application of

probability.

The human mind is notoriously badat estimating probabilities….

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).

Chance of winning the lottery with 6 correct numbers (1-49)is 1 in 13,983,816

Chance of dying in a car accidentis 1.49 in 100

A person is 21,000 times more likely to die in a carthan win the lottery!

Sample Space: All possible outcomes

The sum of all outcomes must be

Exhaustive

and

Mutually exclusive.

Three approaches to probability:

Classical: Probability is as simple as 1/n

Three approaches:

Classical: Probability is as simple as 1/n

Relative Frequency: long –run relative frequency

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

Permutations and Combinations

How many ways can n objectsbe arranged?

n! = n(n-1)(n-2)… etc

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 objectsbe arranged r at a time?

)!(

!

rn

nPnr

How many ways can 4 people be arrangedin unique order in groups of 2?

)!24(

!4

nrP

= 24/2 = 12

Permutations and Combinations

Combination: How many unique groupscan be taken from n objects?

)!(!

!

rnr

nC nr

How many ways can 4 people be arrangedin unique order in groups of 2?

)!24(!2

!4

nrC

= 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 willbe selected first for the interview?

What is the probability that you willbe selected first for the interview?

What is the probability that the fourof you will line up at random, and youare 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?

What is the probability that you and Pat will be the first two interviewed?

What is the probability that you and Pat will be the first two interviewed?

What is the probability that you andPat will be interviewed first, and you willbe interviewed before Pat?

Blaise Pascal 1623 - 1662 Pascal’s Triangle

Binominal Coefficients

The coefficients are simply the combinations of n objectstaken r at a time…

Suppose you wanted to know the probability of getting exactly3 heads in ten tosses of a fair coin?

The probability of a head is p = ½, the probability of a tail is q = ½.

rnrnr qpC

1172.0024,1

1205.5. 7310

3 C

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 utilizesthree independent resume assessors.

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 utilizesthree independent resume assessors.

What is the probability of getting this job if all threeassessors must recommend hiring?

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 utilizesthree independent resume assessors.

What is the probability of getting this job if at least two out of three assessors must recommend hiring?

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 thatjurists are biased against Hispanics. Is this true?

Some Interesting Problems

Gender Discrimination1000 men and 1000 women applied to a university,70% of the men were accepted, but only 40% ofthe women.Did gender discrimination take place?

Yes! In fact…. The university discriminated against men!!

Some Interesting Problems

Gender Discrimination1000 men and 1000 women applied to a university,70% of the men were accepted, but only 40% ofthe women.

There were two programs:One excellent: 200 men 800 women appliedOne mediocre: 800 men 200 women applied

Some Interesting Problems

Gender Discrimination1000 men and 1000 women applied to a university,70% of the men were accepted, but only 40% ofthe women.

There were two programs:One excellent: 200 men 800 women appliedOne mediocre: 800 men 200 women applied

Excellent program: 20% men 25% women accepted

Some Interesting Problems

Gender Discrimination1000 men and 1000 women applied to a university,70% of the men were accepted, but only 40% ofthe women.

There were two programs:One excellent: 200 men 800 women appliedOne mediocre: 800 men 200 women applied

Mediocre program: 82.5% men 100% women accepted

Some Interesting Problems

Gender Discrimination1000 men and 100 women applied to a university,70% of the men were accepted, but only 40% ofthe women.

There were too programs:One excellent: 200 men 800 women appliedOne mediocre: 800 men 200 women applied

In total, 700 men were accepted and only 200 women

Simpson’s Paradox

http://en.wikipedia.org/wiki/Simpson%27s_paradox

Leniency vs Reciprocity

Are employees who work for companies with higher paymore likely to be happy with their jobs?

A study of 500 companies found no relationship between averageemployee pay and average workers’ happiness scores. The reportconcludes that higher paid workers are NOT more happy with their

jobs.

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 ableto see the difference between yellow and white cabs80% 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 inthe city and 80 white cabs. What is theprobability that the witness correctlyidentified the color of the car?

O. J. Simpson Trial

Johnnie Cochran argued that evidence that O.J. beathis wife was irrelevant because only 1 in 1,000wife-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 hadpreviously been beaten?

From : Dr. Michael StarbirdUniversity of Texas

DNA Evidence.

Suppose a murder was committed insouthern California. DNA is found atthe scene. A computer match finds a suspect.

DNA Evidence.

Suppose a murder was committed insouthern California. DNA is found atthe scene. A computer match finds a suspect.

The prosecuting attorney tells thejury that there is only one chancein a million that two people would have the same DNA evidence.

DNA Evidence.

Suppose a murder was committed insouthern California. DNA is found atthe scene. A computer match finds a suspect.Therefore they must convict….

DNA Evidence.

Suppose a murder was committed insouthern California. DNA is found atthe scene. A computer match finds a suspect.

What is the actual probabilitythat they have the realmurderer 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 havinga weapon is falsely accused?

There are about 800 million passenger flights in theUnited Sates per year.

Fifty of these were terrorists trying to get a weapon onan 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 22members. 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 22members. 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

Elections and Selections

Who wins?

VotingMethod

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)

Elections and Selections

Pair-Wise Sequential VotingThe 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 VotingThe 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 Disraeli1804 - 1881

Mark Twain1835 - 1910

“There are three types of lies: lies, damned lies, and statistics.”