Week Four Discussion 2
1. In your own words, describe two main differences between classical and empirical probabilities. The differences between classical and empirical probabilities are that classical assumes that all outcomes are likely to occur, while empirical involves actually physically observing and collecting the information.

2. Gather coins you find around your home or in your pocket or purse. You will need an even number of coins (any denomination) between 16 and 30. You do not need more than that. Put all of the coins in a small bag or container big enough to allow the coins to be shaken around. Shake the bag well and empty the coins onto a table. Tally up how many heads and tails are showing. Do ten repetitions of this experiment, and record your findings every time. * State how many coins you have and present your data in a table or chart. For this experiment, I am using 20 coins.

* Consider just your first count of the tossed coins. What is the observed probability of tossing a head? Of tossing a tail? Show the formula you used and reduce the answer to lowest terms. On my first count of the tossed coins, the probability of heads showing was 10/20=1/2. The probability of tails showing was 10/20=1/2 * Did any of your ten repetitions come out to have exactly the same number of heads and tails? How many times did this happen? Yes and this on happen once, which was on my first roll.

* How come the answers to the step above are not exactly ½ and ½? Actually they are exactly ½ and ½.
* What kind of probability are you using in this “bag of coins” experiment? This experiment was empirical probability because I had to physically observe the information. * Compute the average number of heads from the...

...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...

...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...

...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...

...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...

...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,...

...
a. Suppose Evan chose a bottle from the refrigerator at random. Could we realistically say that the probability of choosing a diet soda is 7/3? Why or why not?
b. If there are 16 total bottles of diet soda, 8 total bottles of regular soda, and 4 total bottles of water, what is the probability of each of the following:
(i) Choosing a bottle of diet soda when a bottle is chosen at random
(ii) Choosing a bottle of regular soda when a bottle is chosen at random
(iii) Choosing a bottle of water when a bottle is chosen at random
Assessment 9-2B: Exercises 1 & 5
Suppose an experiment consists of spinning X and then spinning Y, as follows:
Find the following:
a. The sample space S for the experiment
b. The event A consisting of outcomes from spinning an even number followed by an even number
c. The event B consisting of outcomes from spinning at least one 2
d. The event C consisting of outcomes from spinning exactly one 2
Following are two boxes containing colored and white balls. A ball is drawn at random from box 1. Then a ball is drawn at random from box 2, and the colors of balls from both boxes are recorded in order.
Find each of the following:
a. The probability of two white balls
b. The probability of at least one colored ball
c. The probability of at most one colored ball
d. The probability of...

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