PROBABILITY DISTRIBUTION
In the world of statistics, we are introduced to the concept of probability. On page 146 of our text, it defines probability as "a value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur" (Lind, 2012). When we think about how much this concept pops up within our daily lives, we might be shocked to find the results. Oftentimes, we do not think in these terms, but imagine what the probability of us getting behind the wheel of a car twice a day, Monday through Friday, and arriving at work and home safely. Thankfully, the probability for me has been 'one'! This means that up to this point I have made it to work and returned home every day without getting into an accident. While probability might have one outcome with one set of circumstances, this does not mean it will always turn out that way. Using the same example, just because I have arrived at work every day without getting into an accident, this does not mean it will always be true. As I confess with my words, and pray it does stay the same, probability tells me there is room for a different outcome.

In business, we often look at the probability of success or financial gain when making a decision. There are several things to take into consideration such as the experiment, potential outcomes, and possible events. An experiment is a process that leads to the occurrence of one and only one of several possible observations, while an outcome is a particular result of an experiment (Lind, 2012, p.146). An event is a collection of one or more outcomes of an experiment (Lind, 2012, p.147). There are three different types of probabilities that may be used in business decisions. The first is classical probability, which means that the outcomes of an experiment are equally likely. For example, say a business places an opening for job and the only requirement is the applicants are over the age of 18. Let's assume four people applied,...

...Decision Analysis
Course Outline, Quarter I, 2006
Class Materials Topic
Hardcopy in Packet Other*
Introduction
1 Freemark Abbey Winery Structuring Decisions
Framework for Analyzing Risk
2 The North Star Concert North Star.xls Best Guess, Worst Case, Best Case; and Continuous Uncertainties
3 Engine Services, Inc.
Quick Start Guide to Crystal Ball
Analyzing Uncertainty, Probability Distributions, and Simulation Learning Module: Crystal Ball Litigate Demo
Engine Services.xls Language of Probability Distributions and Monte Carlo Simulation
4 Taurus Telecommunications Corporation: A New Prepaid Phone Card Learning Module: Tornado Sensitivity
Taurus Telecommunications.xls Sensitivity Analysis and Key Drivers
Time Value of Money
5 Dhahran Roads (A)
Evaluating Multiperiod Performance Multiperiod Pro Forma and NPV
6 Roadway Construction Company NPV, IRR, and Project Assumptions
Data and Distributions
7 Appshop, Inc. Simulating NPV
8 Lorex Pharmaceuticals
Introduction to Analytical Probability Distributions Lorex Exhibit 2.xls Distributions
9 Sprigg Lane (A) Sprigg2.xls Probability Distributions and Spreadsheet Modeling; Risk
10 The Waldorf Property
Chapter 11 of QBA: Text and Cases
Waldorf.xls Cumulative Distribution Functions, Adjustment for Risk
11 Amore Frozen Foods (A) Macaroni and Cheese Fill Targets
Sampling Amore.xls Sample Uncertainty
Regression
12 Hightower Department Stores:...

...Probability Theory and Game of Chance
Jingjing Xu
April 24, 2012
I. INTRODUCTION
Probability theory is the mathematical foundation of statistics, and it can be applied to many areas requiring large data analysis. Curiously, that the study on probability theory has its root in parlor games and gambling. In 17th century, dice gambling was a very common entertainment among the upper class. An Italian mathematician and gambler Gerolamo Cardano founded the concept of probability by studying the rules of rolling dice: since a die is a cube with each of its six faces showing a different number from 1 to 6, when it is rolled, the probability of seeing each number is equal. Therefore, some of the gamblers began to wonder, that taking a pair of dice and rolling them a couple of times, which has the larger probability of seeing a sum of 9 or seeing a sum of 10? What about seeing double sixes? In a correspondence between Blaise Pascal and Pierre Fermat, the problems were resolved, and this triggered the first theorem in the modern theory of probability.
II. BASIC DEFINITIONS
Definition 1
In probability theory, the sample space, often denoted Ω, of an experiment is the set of all possible outcomes....

...A Short History of Probability
Dr. Alan M. Polansky
Division of Statistics
Northern Illinois UniversityHistory of Probability 2
French Society in the 1650’s
! Gambling was popular
and fashionable
! Not restricted by law
! As the games became
more complicated and
the stakes became
larger there was a
need for mathematical
methods for computing
chances.History of Probability 3
Enter the Mathematicians
! A well-known gambler,
the chevalier De Mere
consulted Blaise Pascal
in Paris about a some
questions about some
games of chance.
! Pascal began to
correspond with his
friend Pierre Fermat
about these problems.History of Probability 4
Classical Probability
! The correspondence between Pascal and Fermat
is the origin of the mathematical study of
probability.
! The method they developed is now called the
classical approach to computing probabilities.
! The method: Suppose a game has
n equally
likely outcomes, of which
m outcomes
correspond to winning. Then the probability of
winning is m/n.History of Probability 5
Problems with the Classical Method
! The classical method requires a game
to be broken down into equally likely
outcomes.
! It is not always possible to do this.
! It is not always clear when possibilities are
equally likely.History of Probability 6
Experience
!...

...I. Probability Theory
* A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
* The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory ofprobability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples.
* It is the likeliness of an event happening based on all the possible outcomes. The ratio for the probability of an event 'P' occurring is P (event) = number of favorable outcomes divided by number of possible outcomes.
Example:
A coin is tossed on a standard 8×8 chessboard.
What is the theoretical probability that the coin lands on a black square?
Choices:
A. 0.5
B. 0.25
C. 0.42
D. 0.6
Correct answer: A
Solution:
Step 1: Theoretical probability = number of favorable outcomes / number of possible outcomes.
Step 2: The probability of the coin lands on the black square is 32.
Step 3: Total number of outcomes = 64.
Step 4: P (event) =
Step 5: == 0.5
Step 6: The theoretical probability that...

...Hey guys, this is the probability Assignment. Last date for submission is 10 aug...
Q1. What is the probability of picking a card that was either red or black?
Q2. A problem in statistics is given to 5 students A, B, C, D, E. Their chances of solving it are ½,1/3,1/4,1/5,1/6. What is the probability that the problem will be solved?
Q3. A person is known to hit the target in 3 out of 4 shots whereas another person is known to hit the target in 2 out of 3 shots. Find the probability that the target being hit at all when they both try?
Q4. An investment consultant predicts that the odds against price of a certain stock will go up during the next week are 2:1 and the odds in the favor of the price remaining the same are 1:3.What is the probability that the price of the stock will go down during eth next week?
Q5. A bag contains 10 White and 6 Black balls. 4 balls are successfully drawn out and not replaced. What is the probability that they are alternately of different colors?
Q6.In a multiple-choice question there are 4 alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed up to 3 chances to answer the question, find the probability that he will get marks in the question?
Q7. A and B are two independent...

...Notation for the Binomial Distribution
P(S) The symbol for the probability of success
P(F) The symbol for the probability of failure
p The numerical probability of a success
q The numerical probability of a failure
P(S) = p and P(F) = 1 - p = q
n The number of trials
X The number of successes
The probability of a success in a binomial experiment can be computed with the following formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works will be given in the following example.
Example 1:
A coin is tossed three times. Find the probability of getting exactly two heads.
Solution:
This problem can be solved by looking that the sample space. There are three ways to get two heads.
HHH, HHT, HTH, THH, TTH, THT, HTT, TTT
The answer is or 0.375.
The probability of a success in a binomial experiment can be computed with the following formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works will be given in the following example.
Example 1:
A coin is tossed three...

...Hume on Probability
Hume begins section six of “An Enquiry Concerning Human Understanding” by stated right out that chance does not exist, but is merely a result of our ignorance of the causes behind any given event. He argues this by relating probability and belief. Belief arises when probability is at its most high. According to chance, any event may turn out anyway. Hume illustrates his point with a die. If a die were marked with one figure on four sides, while another figure on the other two sides, then it would be most probable that the die would land on the former side. If, however, the die had a thousand sides marked in one manner and only one side marked differently, then the probability of landing on the former mark would be higher. As such, our belief or expectation of this result would be higher1. As the chance of landing on one side of the die increases, the probability of that result also increases, and as such our belief in that result increases. As experience tells us that one result is more probable then another, so our mind construes the belief in that result. The nature of belief is thus constructed, as an experiment is repeated (such as the tossing of the fictional thousand sided die) and the result shows itself to be the same more often then not, then the idea of obtaining that result becomes more concrete and secure in the mind of the observer. Thus constituting a belief. There are...

...Conditional Probability
How to handle Dependent Events
Life is full of random events! You need to get a "feel" for them to be a smart and successful person.
Independent Events
Events can be "Independent", meaning each event is not affected by any other events.
Example: Tossing a coin.
Each toss of a coin is a perfect isolated thing.
What it did in the past will not affect the current toss.
The chance is simply 1-in-2, or 50%, just like ANY toss of the coin.
So each toss is an Independent Event.
Dependent Events
But events can also be "dependent" ... which means they can be affected by previous events ...
Example: Marbles in a Bag
2 blue and 3 red marbles are in a bag.
What are the chances of getting a blue marble?
The chance is 2 in 5
But after taking one out you change the chances!
So the next time:
* if you got a red marble before, then the chance of a blue marble next is 2 in 4
* if you got a blue marble before, then the chance of a blue marble next is 1 in 4
See how the chances change each time? Each event depends on what happened in the previous event, and is called dependent.
That is the kind of thing we will be looking at here.
"Replacement"
Note: if you had replaced the marbles in the bag each time, then the chances would not have changed and the events would be independent:
* With Replacement: the events are Independent (the chances don't change)
* Without Replacement: the events are Dependent (the chances...