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    Probability Theory

         Probability theory is concerned with analyzing random phenomena such as dice rolls, coin flips, and slot machines. It is the basis of the mathematical field of statistics. Probability theory considers both discrete and continuous events. An example of a question concerning discrete events would be: “Heads or tails?” An example of a question concerning continuous events would be: “What time will the train arrive?”

         Probability theory starts by considering a sample space of an event. A sample space is simply the set of all possible outcomes. For example, for a fair die, when rolled, the sample space is {1, 2, 3, 4, 5, 6}. Probability is simply assigning each of these events in the sample space a value between zero and one, with the whole sample space's probabilities summing to one. The probability indicates the likelihood of an event. In this case, each event has the same probability of one sixth.

         Furthermore, the law of large numbers is a theorem that has risen from this basic foundation of probability theory. The law of large numbers simply states that as a sample grows larger and larger, the average result will converge towards the expected outcome of the probability distribution. In this case, it would mean that as you roll the dice more and more times, the running average will approach three and a half.  

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    BrainMass Categories within Probability Theory

    Random Variables

    Solutions: 84

    A random variable is a variable that cannot take on a number of different values subject to chance.

    BrainMass Solutions Available for Instant Download

    Risk game

    You're playing a variant of the board game Risk with your friends. In this variant, the attacker gets to roll two 7-sided dice and the defender rolls a single 8-sided die. The attacker wins if at least one of their dice is strictly higher than the defender's. Find the probability that the attacker wins.

    Binomial Probability Distribution Questions

    1. A fair coin is flipped six times. What is: a. The probability of zero heads b. The probability of 1 heads? c. The probability of 4 heads? d. The expected value of the number of heads? (Show a calculation to verify your answer.) e. The standard deviation of the number of heads? (Show a calculation to verify your answer.)

    Three probability questions

    1. A system contains two components, A and B. The system will function so long as either A or B functions. The probability that A functions is 0.95, the probability that B functions is 0.90, and the probability that both function is 0.88. What is the probability that the system functions? 2. A system contains two components,

    MATH 123: Probability

    Health Tech Gen Studies Total Male 137 686 724 1547 Female 412 172 885 1469 Total 549 858 1609 3016 The above table shows a sample of students by major. Round % answers for 12-17 to one decimal place. 12 What is the probability that a student chosen at random is majoring health? 13 What is the probability that a stud

    Simple Probability Questions

    Consider a container filled with balls in a variety of colors. The bar graph below represents the number of balls of each color in the container. Red Green Yellow Blue Orange Purple Brown 2. If you select one ball randomly from the container (No Peeking!), what is the likelihood that it will be orange? Answer as percent ro

    The Unfair Die

    You are given a loaded six-sided die. Such a die can produce any integer from 1 to 6. The die rolls a 4 three times as often as any other number. What is the probability of getting a 4? You now roll the die 5 times as an experiment. What are the chances of the following scenarios? Instead of solving for event and sample spaces

    Combinations: Selecting Blocks

    Suppose that a box contains 5 RED blocks, 3 BLUE blocks, and 2 GREEN blocks. Suppose that 8 are drawn one after another without replacement. Then WHAT is the EXPECTED number of BLUE blocks drawn?

    The most powerful die

    In a dice game, two players each pick one of three dice and roll them. Whoever rolls the higher number wins (it doesn't matter how much higher). Here are the three dice. A die is considered "more powerful" than another die if it wins more than it loses. For example, if A beats B 70% of the time, then we would say that die A i

    Normal Distribution: Patient Weight

    The distribution of the weight of patients at a local urgent care clinic is normal with a mean of 150 pounds and a standard deviation of 10.6 pounds use this information to answer . What percentage of patients weight less than 130 pounds? What percentage of patients weight at least 160 pounds What percentage of patie

    Probability Questions 3A

    1. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. About 68% of all pregnancies last between a. 250 and 282 days b. 234 and 298 days c. 218 and 314 days d. 250 and 266 days 2.

    Probability Questions 2A

    1. The function p(x) is the probability function for the random variable X. If p(1) = P(X=1) =.2, p(2)= P(X=2) = .1, p(3)=.3, f (7)=.1, and p(10)=.3. What is E(X)? a. 3.4 b. 5.0 c. 9.8 d. 4.2 2. What is bigger C124 or C128? (Note: Cnk is the number of ways of choosing k things out of n.) a. C124

    Probability Questions 1B

    11. A couple has 3 children. What is the probability there are exactly two girls among them. a. 3/8 b. 1/ 2 c. 7/8 d. 5/8 12. Same setup. What is the probability of at most 2 girls. a. 3/8 b. 1/ 2 c. 7/8 5/8 13. If a restaurant offers 3 different appetizers, 5 entrees (main dishes), 4 desserts, an

    Series of Probability Questions

    1. A survey of 120 people found that 20 of them use shampoos A and C, 10 use A and B but not C, 15 use all three, 30 use only C, 35 use B but not C, 25 use B and C, and 10 use none of the three. What percentage of the people surveyed use only A. (Hint: fill in Venn diagram from the inside out - from the intersection of all three

    Combining Probabilities and The Law of Large Numbers

    1. Suppose there are 15 jelly beans in a box 2 red, 3 blue, 4 white, and 6 green. A jelly bean is selected at random. a) What is the probability that the jelly bean is white? _____ b) What is the probability that the jelly bean is not white? _____ c) What is the probability that the jelly bean is green? _____ d) What is the

    Two Conditional Probability Questions

    1 Every Monday, James has a math class and a biology class. The probability that he will have his math homework done is 0.42 and the probability he will have his biology homework done is 0.53. If the probability he will have his biology homework done but not his math homework is 0.29, what is the probability he will have his ma

    Multiple choice computations for a math competition

    Joe takes part in math competitions. A particular contest consists of 25 multiple-choice questions, and each question has 5 possible answers. It awards 6 points for each correct answer, 1.5 points for each answer left blank, and 0 points for incorrect answers. Joe is sure of 12 of his answers. He ruled out 2 choices before g

    Standard Error: Random Coins

    2. Assume that you have 30 coins in a bowl. If you mix them well and grab ten coins, with your eyes closed, thousands of times, what would be the average sample portion of heads? What would be the standard error? 3. If you conducted the experiment in question 2 grabbing only six coins at a time, what would be the average s

    Measurement Scales Identified

    Identify which measurement scale is used in each question and why. 1) For the following list of cities, please rank each on its quality of life (i.e., jobs, taxes, crime rate, employment outlook, citizens' friendliness, cost) which one indicating the best and five the worst. Atlanta, Georgia

    Keys, Marbles and Class

    1) You carry five keys in your pocket, two of which are for the two locks on your front door. You lose one key. What is the probability that you can get into your house through the front door? (Enter your answer as a fraction.) 2) Use the given values to find the following. (Enter your answers as fractions.) P(A) = 0.6,

    Probability: Cholesterol, Smoking and Hyperlipidemia

    1. The following box-whisker plot shows the distributions of total cholesterol levels in boys and girls 10-15 years of age. (see the attachment for the box-whisker plot) A. What is the median total cholesterol level in boys? ________ B. What is Q1 for boys? ________ C. What is Q3 for boys? ________ D. Lower Limit

    Probability, Combinations and Permutations

    Find the indicated probability 1) The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker, Occasional smoker, Regular smoker, Heavy smoker, Total Men, 431, 50, 71, 49, 601 Women, 382, 48, 86, 39, 555 Total, 813, 98, 157, 88, 1156 If one of the 1156 people is randomly selected. find the prob

    Binominal Probability Concepts and Distributions

    Probability Concepts and Distributions: [6] In a clinical trial of Lipitor, a common drug used to lower cholesterol, 863 patients were given a treatment of 10-mg Atorvastatin tablets. Among them, 19 patients experienced flu symptoms and 844 patients did not (based on data from Pfizer, Inc.). a) What would you think that the m

    Probability: black-and-white and color copiers

    A local FedEx/Kinkos has three black-and-white copy machines and two color copiers. Based on historical data, the chances that each black-and-white copier will be down for repairs is 0.10. The color copiers are more of a problem and are down 20% of the time each. a. Based on this information, what is the probability that if

    Probabilities of Poker Hands

    See the following link on Poker Probability: http://en.wikipedia.org/wiki/Poker_probability Find the probability of 13 cards out of a deck of poker card Find the following odds shown on the link. The odds of a royal flush, straight flush, four of kind, full house, flush, straight, three of kind, 2 royal flush, 2 straight

    Conditional Probabilities on Hidden Prizes

    Need help with calculating probability. I have tried to solve the problems below, but I need you to give me step-by-step instructions to make sure that I understand how to do the problems correctly. 1. Use the information below to answer Items 10-11: A game has three boxes. Box 1 has one drawer, Box 2 has two drawers, and B

    Calculate probability under either binomial or normal distribution

    All problems are to be computed by hand with work shown; type your work up in word. Question 1: The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with a mean of 266 days and standard deviation of 16 days. •What percent of pregnancies last less than 240

    Probability and the Win 4 Lottery

    In the New York State Win 4 lottery, you place a bet by selecting four digits. Repetition is allowed, and winning requires that your sequence of four digits matches the four digits that are later drawn. Assume you placed one bet with a sequence of four digits. a. Use the multiplication rule to find the probability that your f

    Testing of hypothesis - logical questions

    Need clarification regarding the questions below - thank you in advance. 1. The manager of the quality department for a tire manufacturing company wants to know the average tensile strength of rubber used in making a certain brand of radial tire. The population is normally distributed and the population standard deviation is

    Inclusion-exclusion principle

    The two problems are taken from this derivation of the inclusion-exclusion principle: http://www.proofwiki.org/wiki/Inclusion-Exclusion_Principle#Induction_Hypothesis They are added in the document. Let me know if the documents does not work

    Normal z score for normally distributed data

    Four students each received a raw score of 22 on a different test. Compute the z-score for the raw score value of 22 for each student, given information about the distributions of X values for each test. Which student (a, b, c, or d) had the highest score value, compared to the normative group who took the same test? Which stude