12.3 We use equation 2 to find out probability:
F(t)=1 – e^-Lt
1-e^-(0.4167)(10) = 0.98 almost certainty. This shows that probability of another arrival in the next 10 minutes. Now we figure out how many customers actually arrive within those 10 minutes. If the mean is 0.4167, then 0.4167*10=4.2, and we can round that to 4.

X-axis represents minutes (0-10)
Y-axis represents number of people.
We can conclude from this chart that the highest point with the most visitors is in the beginning of the 10 minutes. There may be a dispersion of visitors between the times, which according to this would be the slowest times. We can see 1 customer also visiting at the end of the 10 minutes. If a curve was to be drawn on this graph, it would signify a decline in visitors from point 0 to 6 and a steady move from 6 to 10.

16.3

The Lower- Colorado River Authority (LCRA) has been studying congestion at the boat-launching ramp near Mansfield Dam. On weekends, the arrival rate averages 5 boaters per hour, Poisson distributed. The average time to launch or retrieve a boat is 10 minutes, with negative exponential distribution. Assume that only one boat can be launched or retrieved at a time. a.)The LCRA plans to add another ramp when the average turnaround time exceeds 90 minutes. At what average arrival rate per hour should the LCRA begin to consider adding another ramp? b.)If there were room to park only two boats at the top of the ramp in preparation for launching, how often would an arrival find insufficient parking space?

a.) The number of arrivals is Poisson distributed with mean 5 per hour. The service time is exponential distributed with mean service time 10 minutes or the number of services is Poisson distributed with mean 1hr/10minutes = 60/10= 6 meaning only 1 server. Arrival rate = L = 5 boats per hour

Inter arrival time =1/L= 1/5 = 12minutes
Average service time = 1/u= 10 minutes= 1/6 hours
Service rate =u = 6 boats per hour...

...chapter, you will be able to ONEDefine probability. TWO Describe the classical, empirical, and subjective approaches to probability. THREEUnderstand the terms experiment, event, outcome, permutation, and combination. FOURDefine the terms conditional probability and joint probability. FIVE Calculate probabilities applying the rules of addition and multiplication. SIXUse a tree diagram to organize and computeprobabilities. SEVEN Calculate a probability using Bayes theorem. What is probability There is really no answer to this question. Some people think of it as limiting frequency. That is, to say that the probability of getting heads when a coin is tossed means that, if the coin is tossed many times, it is likely to come down heads about half the time. But if you toss a coin 1000 times, you are not likely to get exactly 500 heads. You wouldnt be surprised to get only 495. But what about 450, or 100 Some people would say that you can work out probability by physical arguments, like the one we used for a fair coin. But this argument doesnt work in all cases, and it doesnt explain what probability means. Some people say it is subjective. You say that the probability of heads in a coin toss is 12 because you have no reason for thinking either heads or tails more likely you might change your view if you knew...

...I. ProbabilityTheory
* 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 of probability. 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...

...random variable is
A) generated by a random number table.
B) the variable for which an algebraic equation is solved.
C) a numerical measure of a probability experiment.. Ans = C
D) a qualitative attribute of a population.
4) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer yes or no.
x 5 10 15 25
P(x) 0.1 –0.1 0.3 0.8 Ans = No
5) True or False: The expected value of a discrete random variable may be negative Ans = True
6) The table of probabilities of the random variable x is given as:
x 0 1 2 5
P(x) 0.5 0.2 0.2 0.1
Find the mean, µ and standard deviation, σ of x. Round answers to one decimal place. Ans = µ = 1.1, σ = 1.5
7) If p is the probability of success of a binomial experiment then the probability of failure is
A) 1 B) –p C) 1–p D) p + 0.5 Ans = C
8) A binomial experiment has 6 trials with the probability of success on any trial = p = 0.5. Find the probability of exactly 2 successes in the 6 trials. (Use the binomial probability distribution function.) Ans = 0.2344
9) Assume that male and female births are equally likely and the birth of any child does not affect the probability of the gender of any other children. Find the...

...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...

...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...

...susceptibility to the disease?
2. If the contagion rate is calculated as the number of new cases per day per total population, what would the average contagion rate be for Kold?
Unlike some of the other interactive labs, this model has some randomness built in to reflect the real spread of a disease, which is a matter of probabilities. Despite this variability, you can get a sense for what effect each factor has on disease spread.
Before running the simulator, predict whether the sick days per capita will be higher or lower with low population density.
Record your prediction in the data table and then run the simulator to 100 days three times, recording the data each time.
Make a prediction for high population density; record it in the data table, and run the simulator three times, recording that data in the table. Answer the following:
3. What could be done to prevent the spread of disease in a low population density?
4. What kinds of challenges would high population density present to these precautions?
5. If contagion rate is calculated as the number of new cases per day per total population, what would the average contagion rate be for Kold?
Population mixing in a contagious area is analogous to increasing population density. Both increased density and increased movement of people bring more contagious people into contact with susceptible people,...

...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:...

...ProbabilityTheory and Game of Chance
Jingjing Xu
April 24, 2012
I. INTRODUCTION
Probabilitytheory is the mathematical foundation of statistics, and it can be applied to many areas requiring large data analysis. Curiously, that the study on probabilitytheory 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 probabilitytheory, the sample space,...

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