...Shamara MAT 143 03/02/16 Module 6 Project 1. (5 pts) You are going to roll a pair of dice 108 times and record the sum of each roll. Before beginning, make a prediction about how you think the sums will be distributed. (Each sum will occur equally often, there will be more 12s than any other sum, there will be more 5s than any other sum, etc.) Record your prediction here: I think in the beginning the sums will vary but then eventually repeat, and there will be more 7’s then any other sum. 2. (5 pts) Roll the dice 108 times and record the sum of each roll in the table provided below. 7 3 4 8 11 3 6 4 2 10 5 9 7 7 12 10 10 3 5 2 2 6 4 8 5 9 2 7 9 7 7 12 12 4 6 3 9 8 2 3 12 6 9 7 7 7 12 5 6 8 2 4 7 7 8 10 3 5 5 4 7 7 12 4 6 8 3 12 7 7 8 3 10 3 5 8 7 9 6 6 12 4 7 9 9 11 4 5 2 5 2 3 5 3 10 2 2 4 3 6 3 10 7 7 9 8 5 12 3. (22 pts) Find the experimental probability of rolling each sum. Fill out the following table: Sum of the dice Number of times each sum occurred Probability of occurrence for each sum out of your 108 total rolls (record your probabilities to three decimal places) 2 10 .092 3 13 .120 4 10 .092 5 11 .101 6 9 .083 7 19 .175 8 9 .083 9 9 .083 10 7 .064 11 2 .018 12 9 .083 4. (5 pts) Compare your outcomes to your prediction. Was your prediction correct? Why do you think this happened? Yes, my predictions were correct and I think it just happened coincidently. 5. (5 pts) What number(s) occurred...
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...following results. The probability of getting a possible 3 if we roll one die, is 1/6 as the die has 6 sides and the number 3 is only in 1 side of the die. Although, if we roll two dice, the probability of getting a possible 3 in at least one die, is twice as likely to occur with a chance of 1/6 in the one die plus 1/6 on the other, which is equal to 2/6 for both dice and therefore simplified to 1/3. In the meantime, there are six conditions that the die A might get a 3 and six conditions that die B might be the same number as die A; thus one condition in both dice that they get a possible 3. The probability for the example explained is 12/36 minus 1/36, which is equal to 11/36....
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...Title: The Probability that the Sum of two dice when thrown is equal to seven Purpose of Project * To carry out simple experiments to determine the probability that the sum of two dice when thrown is equal to seven. Variables * Independent- sum * Dependent- number of throws * Controlled- Cloth covered table top. Method of data collection 1. Two ordinary six-faced gaming dice was thrown 100 times using three different method which can be shown below. i. The dice was held in the palm of the hand and shaken around a few times before it was thrown onto a cloth covered table top. ii. The dice was placed into a Styrofoam cup and shaken around few times before it was thrown on a cloth covered table top. iii. The dice was placed into a glass and shaken around a few times before it was thrown onto a cloth covered table top. 2. All result was recoded and tabulated. 3. A probability tree was drawn. Presentation of Data Throw by hand Sum of two dice | Frequency | 23456789101112 | 4485161516121172 | Throw by Styrofoam cup Sum of two dice | Frequency | 23456789101112 | 2513112081481072 | Throw by Glass Sum of two dice | Frequency | 23456789101112 | 18910121214121174 | Sum oftwo dice | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total | Experiment1 | 4 | 4 | 8 | 5 | 16 | 15 | 16 | 12 | 11 | 7 | 2 | 100 | Experiment2 | 2 | 5 | 13 | 11 | 20 | 8 | 14 | 8 | 10 | 7 | 2 | 100 | Experiment3 | 1 | 8 | 9 | 10 | 12 | 12 | 13 | 12 | 11...
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...Lab I – Probability Models When finished fill in the table at the end of this document and email it to your teacher Cards: Use the tab for Cards in the excel file and get the total probability for the various hands All of your inputs are on the yellow cells (a) The first task is to figure how many chances you have for each pick that will result in a successful hand INPUT: columns B/C/D/E/F (b) Then if any of the events (one of the five cards) has 100% chance then you need multiply the one path by five to account for the parallel paths to a successful hand INPUT: column H (c) Finally, if there are more than one way to get the result (example four of a kind can be done with 13 different cards) then you need to add that number too INPUT: column J Note: this step will be more complicated when dealing with royal hands, and multiple runs. (d) Your final probability will be the percentage in the last column in the “prob” row (I’ve formattesd it such that these all fall in rows 7,17,27, etc) Dice: Use the tab for Dice in the excel file and get the probabilities for all the outcomes when tossing three dice Note: this problem has been started with the first result for the outcomes of 3 and 4 The probability also has been simplified to two events (rather that 3) by using the results from lab H for the rolls for dice #2 and #3… see below (also on your excel file) Outcome | theoretical % | total | possible outcomes (die 1 / die 2) | 2 |...
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...Contents No. | Title | Page | 1 | Introduction | 2 | 2 | Part 1 | 6 | 3 | Part 2 | 8 | 4 | Part 3 | 10 | 5 | Part 4 | 13 | 6 | Part 5 | 17 | 7 | Further Exploration | 21 | 8 | Reflection | 22 | Introduction Moral Values: I have learned many moral values while completing this assignment. Better still, I got to know about the importance of applying these moral values into our daily lives. The first moral value is to cooperate with other people. The strategies and solutions of the questions in this project work were discussed among me and a group of friends. This makes things easier and saved a lot of time. The management of time is also important to complete this project work. Other than this assignment, I have homework, extra co curricular activities and tuition classes to attend. Thus a good management of time is essential for me to complete this given task and not to disrupt my daily activities. Perseverance has taught me to be steady and persistent in doing something. In spite of many difficulties I faced throughout the whole procedure I learned that giving up is just not right solution. Obstacles and discouragement should be endured the course of action should be held on with unyielding determination to see obtain sweet fruit of success. I had also learned to appreciate the beauty of mathematics. Waxing eloquently on the basic importance of Mathematics in human life, Roger Bacon (1214-1294), an English Franciscan friar, philosopher, scientist...
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...Standard 7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. It is Fall time and what does that mean, high school football. There are 11 teams in your region each of them are equal matches to one another. It really is the luck of the...
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...Easy Probability Problems 1 A class consists of 13 boys and 17 girls. If a student is chosen at random from the class, find the probability that the student is (i) a girl, (ii) a boy 2 A card is drawn from a full pack (with no jokers). Find the probability that the card chosen is (i) a red card, (ii) a diamond, (iii) a spade, (iv) the ace of clubs. 3 A letter is chosen at random from the word PARALLEL. Find the probability that the letter chosen is (i) an L, (ii) a vowel, (iii) a consonant. 4 The probability of a potato being bad is 0.1. A bag contains 110 potatoes. How many good potatoes would you expect to find in the bag? 5 A die is rolled. Find the probability of getting (i) a five, (ii) a five or a six, (iii) a number less than four. 6 Two dice are rolled. Find the probability of getting (i) a six on both dice, (ii) a total of 9 on the two dice, (iii) a total of over ten on the two dice. 7 A pair of dice is rolled 180 times during a game of Monopoly. How many times would you expect (i) a total of 7, (ii) a total of 4, (iii) a total of 2. 8 A classroom contains 14 girls and 16 boys. A student is picked at random and leaves the room. Another student is now picked at random. Find the probability that (i) both are boys, (ii) both are girls, (iii) at least one is a girl 9 A card is drawn at random from a pack but NOT replaced. Then, another card is drawn from the pack. Find the probability that (i) the first is a Jack and the second a...
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...distribution. a. Throw a fair dice once. What is the sample space? What is the probability to get “six”? What is the probability to get “six” or “five”? Define variable X to be { 1,2,……6} Probability to get a 6 is P(six)=P(X=6) = 1/6 Probability to get a 5 or 6 is 1/3 The sample space is the set of all possible outcomes. Sample space = {(one)(two)…(six)} b. Throw a fair dice 10 times. What is the probability to get “six” twice? What is the probability to get six at least twice? Define random variable X to be number of getting “six” (successs). P(X=2) c. Throw a fair dice 10 times. What is the expected value and variance of getting “six”? E(X)= n*p=10*1/6=1.67 Var(X)=n*p=(1-p)=1.39 d. If you throw the dice 100 times, how does the probability distribution of getting “six” look like? 2. Poisson distribution. Suppose we know that the average arrival of cars per hour at a gas station is 15. a. What is the probability that only 5 cars have arrived in one hour? Define r.v X to be … P(X=5)= 1.002 b. What is the expected value and variance? E(X)= 15=Var(X) … c. How does the probability distribution look like? … 3. Normal distribution (continuous distribution) Suppose the monthly return of S&P 500 index follows normal distribution with μ=0, σ=0.05. a. What is the probability that the return of next month is at least 2%? Define X to be … P(X>=2%) = 1-P(X<2%)= .034 b. What is the probability that the return of next month...
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...There are two types of distinct interpretations of the term probability which is to determine the probability of a particular outcome and observing the relative frequency over many repetitions of the situations. The first method is to make assumptions about the physical world. For example assume that dice are made in a way such that they are likely to land on any of their six surfaces and that the way in which I throw one dice has no effect on how it lands. The probability of the number six showing on top is one out of six. The second method is if I would have to stand there and throw the dice several different times and based on how it lands can determine the probability. Making a determination on the least amount I throw the dice will have a bad outcome Davis, (2001). My decision to use the throw of dice in this example is not probabilistic but choosing a very specific reason in describing the way in which I have heard about business decisions made by companies. The level of technology as a service offering to its customers and public ownership of the company are of a stable influence in making decisions based on short and shallow analysis of their probability for success. You can see this on the news almost every day or by reading The Wall Street Journal. In this economy is driven by expense, so much more in public owned companies where shared of resources to research and investigation are often recognized as pointless by stockholders. My point is that if you can not invest...
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...Probability Discrete Event Simulation: Tutorial on Probability and Random Variables 1. A card is drawn from an ordinary deck of 52 playing cards. Find the probability that it is (a) an Ace, (b) a jack of hearts, (c) a three of clubs or a six of diamonds, (d) a heart, (e) any suit except hearts, (f) a ten of spade, (g) neither a four nor a club (1/3, 1/52, 1/26, 1/4, 3/4, 4/13, 9/13) 2. A ball is drawn at random from a box containing six red balls, 4 white balls and 5 blue balls. Determine the probability that it is (a) red, (b) white, (c) blue, (d) not red, (e) red or white (2/5, 4/15, 1/3, 3/5, 2/3) 3. Three balls are drawn successively from the box in the previous problem. Find the probability that they are drawn in the order red, white and blue if each ball is (a) replaced (b) not replaced (8/225, 4/91) 4. A fair die is tossed twice. Find the probability of getting a 4, 5 or 6 on the first toss and a 1, 2, 3, or 4 on the second toss. (1/3) 5. Find the probability of not getting a 7 or 11 total on either of two tosses of a pair of fair dice. (49/81) 6. Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability that they are both aces if the first card is (a) replaced, (b) not replaced. (1/169, 1/221) 7. Box I contains 3 red and 2 blue marbles while box II contains 2 red marbles and 8 blue marbles. A fair coin is tossed. If the coin turns up heads a marble is chosen from box I and if it turns up tails, a marble is...
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...What is the sample space? (b) What is the probability that the first ball drawn is red? (c) What is the probability that at least one of the two balls drawn is red? (d) What is the (conditional) probability that the second ball drawn is red, given that the first ball drawn is red? Solution. (a) The experiment is the drawing of two balls from the urn without replacement. The sample space is the set of possible outcomes, of which there are four: drawing two red balls; drawing two yellow balls; drawing a red ball first, and then a yellow ball; and drawing a yellow ball first, and then a red ball. One way to denote the sample space is in set notation, abbreviating the colors red and yellow: sample space = {RR, YY, RY, YR}. Note that these four outcomes are not equally likely. We can also represent the experiment and the possible outcomes in a probability tree diagram, as shown below. Note in particular the probabilities given for the second ball. For example, if the first ball is red, then four out of the remaining seven balls are red, so the probability that the second ball is red is 4/7 (and the probability that it is yellow is 3/7). On the other hand, if the first ball is yellow, then five out of the remaining seven balls are red, so the probability that the second ball is red is 5/7 (and the probability that it is yellow is 2/7). The probabilities of each of the four outcomes can be computed by multiplying the probabilities along the branches leading to the outcomes...
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...Massachusetts Institute of Technology 6.042J/18.062J, Fall ’02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Course Notes 10 November 4 revised November 6, 2002, 572 minutes Introduction to Probability 1 Probability Probability will be the topic for the rest of the term. Probability is one of the most important subjects in Mathematics and Computer Science. Most upper level Computer Science courses require probability in some form, especially in analysis of algorithms and data structures, but also in information theory, cryptography, control and systems theory, network design, artificial intelligence, and game theory. Probability also plays a key role in fields such as Physics, Biology, Economics and Medicine. There is a close relationship between Counting/Combinatorics and Probability. In many cases, the probability of an event is simply the fraction of possible outcomes that make up the event. So many of the rules we developed for finding the cardinality of finite sets carry over to Probability Theory. For example, we’ll apply an Inclusion-Exclusion principle for probabilities in some examples below. In principle, probability boils down to a few simple rules, but it remains a tricky subject because these rules often lead unintuitive conclusions. Using “common sense” reasoning about probabilistic questions is notoriously unreliable, as we’ll illustrate with many real-life examples. This reading is longer than usual . To keep things in bounds...
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...Probability Ravindra S. Gokhale IIM Indore 1 “It is a truth very certain that, when it is not in our power to determine what is true, we ought to follow what is most probable” - Rene Descartes 2 Probability Models Concerned with the study of random (or chance) phenomena Random experiment: An experiment that can result in different outcomes, even though it is repeated in the same manner every time Managerial Implication: Due to this random nature additional capacities for any setup are required Although random, certain statistical regularities are to be captured to form the model Model: An abstraction of real world problem 3 Sample Space A set of all possible outcomes of a random experiment Denoted by ‘S’ Examples: • • • • • • • Tossing of a coin: S = {‘Head’, ‘Tail’} Throwing of a dice: S = {1, 2, 3, 4, 5, 6} Marks of a student in a 20 mark paper with only integer negative marking possible: S = {-20, -19, …, -1, 0, 1, …, 19, 20} Number of people arriving at a bank in a day: S = {0, 1, 2, …} Inspection of parts till one defective part is found: S = {d, gd, ggd, gggd, …} Temperature of a place with a knowledge that it ranges between 10 degrees and 50 degrees: S = {any value between 10 to 50} Speed of a train at a given time, with no other additional information: S = {any value between 0 to infinity} 4 Sample Space (cont…) Discrete sample space: One that contains either finite or countable infinite set of outcomes ...
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...2 Probability To most people, probability is a loosely defined term employed in everyday’s conversation to indicate the measure of one’s belief in the occurrence of a future event. Say, what is the chance of arriving on time if one takes the bus? takes the taxi? However, for scientific purposes, it is necessary to give the word probability a definitive, clear interpretation. One can think of probability as the language in discussing uncertainty. For instance, in tossing a coin (random experiment), the outcomes can either be {H} or {T }. If we know exactly what the outcome of the next trial will be, then we are certain that, say, a {H} will show up next. However, it is impossible to know exactly what is going to happen in reality - uncertainty. Definition. A random experiment is a process leading to at least two possible outcomes with uncertainty as to which will occur. Then, the possible outcomes of a random experiment are called the basic outcomes, and the set of all basic outcomes is called the sample space, S. Examples of random experiment (i) Tossing a coin Basic outcomes: {H}, {T } Sample space, S = {H, T } (ii) Rolling a dice Basic outcomes: {1}, {2}, {3}, {4}, {5}, {6} Sample space, S = {1, 2, 3, 4, 5, 6} (iii) Daily changes in Hang Sang Index Basic outcomes: {up}, {down}, {no change} Sample space, S = {up, down, no change} Definition. An event is a set of basic outcomes, or a collection of some basic outcomes from the sample space, and it is said to occur if the random experiment...
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...Probability: Introduction to Basic Concept Uncertainty pervades all aspects of human endeavor. Probability is one of our most important conceptual tools because we use it to assess degrees of uncertainty and thereby to reduce risk. Whether or not one has had formal instruction in this topic, s/he is already familiar with the concept of probability since it pervades almost all aspects of our lives. With out consciously realizing it many of our decisions are based on probability. For example, when you study for an examination, you concentrate more on areas that you feel are likely to be covered on the test. You may cancel or postpone an out door activity if you believe the likelihood of rain is high. In business, probability plays a key role in decision-making. The owner of a retail shoe store, for example, orders heavily in those sizes that s/he believes likely to sell fast. The owner of a movie theatre schedules matinees only during holiday seasons because the chances of filling the theatre are greater at that time. The two companies decide to merge when they believe the probability of success is greater for the consolidated company than for either independently. Some important Definitions: Experiment: Experiment is an act that can be repeated under given conditions. Usually, the exact result of the experiment cannot be predicted with certainly. Unit experiment is known as trial. This means that trial is a special case of experiment. Experiment may be a trial...
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