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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.

## BrainMass Categories within Probability Theory

### 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:  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

### 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

### Measurements

7 6 9 11 8 9 11 9 10 8 7 7 5 9 10 7 7 7 7 9 12 10 10 8 6 a. compute x (with line over), s^2, and s for this sample b. count the number of measurements in the intervals x (with line over) + and - s, x (with line over) + and - 2s, x (with line over) + and - 3s. Express each count as a percent

### 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

### compute and interpret the range, and the mean deviation

Using the age of people compute and interpret the range, the mean deviation, the variance, and the standard deviation of the ungrouped age data. 2. Group the data into classes with five-year class widths, compute and interpret the range, the variance, and the standard deviation of grouped data. 3. Explain the characteristics

### Active filter types

a) What is an active filter? What are its types? b) Derive the magnitude and the phase of the system gain of the low pass active filter with a pass-band gain Apb = 10 dB, RF = 10 kOhm and Rs = 1.1 kOhm. c) Plot the magnitude and phase frequency response curve for the low-pass active filter of (b).

### Chebyshev's Inequality

Household income does not tend to follow a normal distribution in a particular state, yet average income is approximately \$45,000/year in this state, with a standard deviation of about \$9000. At least what percentage of household incomes in this state is likely to be between \$18,000 to \$72,000/year? Show your work as to how this

### Poisson Distributions, Probabilities for insurance claim filed; 'January theory' for stocks

1.The sales of Lexus automobiles in the Detroit area follow a Poisson distribution with a mean of 3 per day. What is the probability that for five consecutive days at least one Lexus is sold? 2. An auditor for Health Maintenance Services of Georgia reports 40 percent of policyholders 55 years or older submit a claim during