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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.  

Categories within Probability Theory

Random Variables

Postings: 81

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

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

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

Arithmetic mean, median and probablity

1. A property of concern for any food company that uses a high-speed carton-filling machine to package juice is the weight of the food product in the individual cartons. If the cartons are under filled, two problems arise. First, customers may not have enough product for their needs. Second, the company may be in violation of

Probability Problem with Playing Cards

A standard pack of 52 playing cards is shuffled thoroughly and then cut. The pack is then shuffled and cut for a second time. Within the pack, a 'picture' card is defined to be a card showing an ace, king, queen or jack (that is, not a card showing any of the numbers 2, 3,... , 10). ** Please see the attachment for the full

Discrete Least Squares

Suppose you are given the data points x = {x0,x1,x2,......xn} ^T and the function values f= {f0, f1,f2,........fn}^T, where xi > 0 for all i = 0,1,2,......n a) For some reason, you think that h(x)= a + b*x + c*e^(arccos(x)) + d*sin(cos(T23(x))), where T23(x) is the 23rd degree Chebyshev polynomial is a great model for the dat

Statistic Questions on Samples

W2A3 Sample questions 1. For a particular sample of 63 scores on a psychology exam, the following results were obtained. First quartile = 57 Third quartile = 72 Standard deviation = 9 Range = 52 Mean = 72 Median = 68 Mode = 70 Midrange = 57 Answer each of the following: I. What score was earned by more stude

Using Chebyshev's Theorem and Other Statistic Rules

1. I. Use Chebyshev's theorem to find what percent of the values will fall between 162 and 292 for a data set with a mean of 227 and standard deviation of 13. II. Use the Empirical Rule to find what two values 99.7% of the data will fall between for a data set with a mean of 246 and standard deviation of 16. 2. Nine coll

Solving Probability Theory Questions

Grindit & Floggit Ltd. make customized Widgets at a cost of $150 per unit of which 3% (on average) happen to be out of tolerance. Up to 115 Widgets can be produces in one continuous run whose set up incurs an additional cost of $10 000. Since manufacturing runs are set to produce one of many variants of Widget, Grindit & Floggit

Chebyshev's Theorem vs. Empirical Rule

In the following example, why would Chebyshev's Theorem be used instead of the Empirical Rule? The Empirical Rule is a rule in statistics that says for a normal distribution, most of all of the data will land between three standardized yet different deviations from their mean. What the empirical rule does is it displays that

Statistics (The Empirical Rule vs. Chebyshev's Theorem)

How does the Empirical Rule work and how does it relate to the bell curve as illustrated in Figure 3.14 (a)? Then, explain Chebyshev's Theorem and how it is different from the Empirical Rule. Give a specific example of a population with which the Empirical Rule might be most effective and one with which Chebyshev's Theorem might

Cherbyshev's Theorem, Empirical Rule, and Other Statistics Problems

I am having difficulties with these problems. Please help. 1. a) Use Chebyshev's theorem to find what percent of the values will fall between 220 and 316 for a data set with a mean of 268 and standard deviation of 12. b) Use the Empirical Rule to find what two values 99.7% of the data will fall between for a data set with

Chebyshev's Theorem for a Sample of Scores on a Psychology Exam

For a particular sample of 77 scores on a psychology exam, the following results were obtained. First quartile = 44 Third quartile = 71 Standard deviation = 6 Range = 45 Mean = 64 Median = 57 Mode = 92 Midrange = 60 I. What score was earned by more students than any other score? Why? II. What was the highest score

Statistics Theory True-or-False Questions

True or False? 1. Statistics is a tool for turning data into information for making decisions. 2. Counting and categorizing are basic forms of statistics. 3. The mean and the median are essentially the same measures and are interchangeable. 4. Samples with the same means always have the same standard deviations. 5. The c

Probability Theory

1. There are two events A and B, both with nonzero probabilities. If the occurrence of B makes occurrence of A more likely, is it ALWAYS true that the occurrence of A also makes occurrence of B more likely? (hint: "B makes occurrence of A more likely" can be represented as "P(A│B) > P(A)") 2. A parking lot consists o

Price of Gas, Hours of Student Study and UPS Box Weights

See the attached file for the full problems. 1. NBC TV news, in a segment on the price of gasoline, reported last evening that the mean price nationwide is $1.50 per gallon for self-serve regular unleaded. A random sample of 35 stations in the Milwaukee, WI, area revealed that the mean price was $1.52 per gallon and that the

Probability Theory

Given the Probability Axioms: Nonnegativity: P(A)  0 for every event A Additivity: if all Ai are disjoint, then: P(A1  A2 ...) = P(A1) + P(A2) + ... Normalization: P() = 1 1. Let A and B be independent events with AC denoting the complement of A. Prove that AC is independent of B

Probability Theory

In a sample of 1,000 representing a survey from the entire population, 650 people were from Laketown, and the rest of the people were from River City. Out of the sample, 19 people had some form of cancer. Thirteen of these people were from Laketown. Are the events of living in Laketown and having some sort of cancer independe

Standard Deviation

Heights of women have a belt-shaped distribution with a mean of 158 cm and a standard deviation of 7 cm. Using Chebyshev's theorem, what do we know about the percentage of women with heights that are within 2 standard deviations of the mean? What are the minimum and maximum heights that are within 2 standard deviations of the

Basic Statistics

Part I T/F and Multiple Choice Questions 1. Each set of data has four quartiles; they divide the ranked data into four equal quarters. _____ (T/F) 2. For any distribution, the sum of the deviations from the mean equals zero. _____ (T/F) 3. In a data set, the mode will always be unique. _____ (T/F) 4. The mean, m

Probability Theory with Coins, Dice and Cards

1. Suppose you have 2 nickels, 3 dimes, and 8 quarters in your pocket. If you draw a coin randomly from your pocket, what is the probability that a. You will draw a dime? b. You will draw a nickel? c. You will draw a quarter? 2. You are rolling a pair of dice, one red and one green. What is the probability of the

Probability and Statistics

Evaluate the permutation. 1) P( 9, 5) 1) _______ A) 9 B) 1 C) 504 D) 15,120 A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability. 2) All cherry 2) _______ A) .7272 B) .3636 C) .1091 D) .1212 In a ce

Discussion Question

Part A: What is dispersion? Briefly name five measures of dispersion. Describe the characteristics of the standard deviation. Explain! Part B: Briefly describe to what kind of data does Chebyshev's Theorem apply? To what kind of data does Empirical Rule apply? Explain.