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# Probability

### Probability of passengers showing up

Airline overbooking is a common practice. Due to uncertain plans, many people cancel at the last minute or simply fail to show up. Air Eagle is a small commuter airline. Its past records indicate that 80% of the people who make a reservation will show up for the flight. The other 20% do not show up. Air Eagle decided to book 12

### Probability of Random Draws

A box contains 10 chips. The chips are numbered 1 through 10. Otherwise, the chips are identical. From this box, we draw one chip at random, and record its value. We then put the chip back in the box. We repeat this process two more times, making three draws in total from this box. 1. How many elements are in the sample spac

### Probability with a coin toss

A coin is tossed 4 times. Let "A" be the event that the first toss is heads. Let "B" be the event that the third toss is heads. 1. What is the probability that the third toss is heads, given that the first toss is heads? 2. Are "A" and "B" independent? Why or why not?

### Poisson Distribution Probability Problem

During the period of time that a local university takes phone-in registrations, calls come in at the rate of one every two minutes. [Hint: It is a Poisson Distribution Problem.] a) What is the expected number of calls per hour? b) What is the probability of three calls in five minutes? c) What is the proba

### Develop Probability Distribution

A technician services machines as companies in the Phoenix area. Depending on the type of malfunction, the service call can take exactly 1, 2, 3, or 4 hours. The different types of malfunctions occur at the same frequency. a. Develop a probability distribution for the duration of a service call. Duration of Call (x) f(x

### Probabilities and Confidence

1. An important issue facing Americans is the large number of medical malpractice lawsuits and the expenses that they generate. In a study of 1228 randomly selected medical malpractice lawsuits, it is found that 856 of there were later dropped or dismissed (based on data from the Physician Insurers Association of America). a)

### Calculating mean profit and probability using simulation

Develop a worksheet simulation for the following problem. The management of Madeira Manufacturing Company is considering the introduction of a new product. The fixed cost to begin the production of the product is \$30,000. The variable cost for the product is uniformly distributed between \$16 and \$24 per unit. The product will se

### Quality Assurance and Binomial Distribution

John Rengel is the Quality Assurance Supervisor for Vino Supremo Vinyards. He knows that 10 percent of each box of corks is undersized. a) If he were to randomly select 120 corks from the next box, then how many of these corks would John expect to be undersized? b) If he were to randomly select 120 corks from each box,

### Probability Distributions Problem

Determine whether each of the distributions given below represents a probability distribution. Justify your answer so I am able to see how this was done. :) (A) x 1 2 3 4 P(x) 1/12 5/12 1/3 1/12 (B) x 3 6 8 P(x) 2/10 .5 1/5 (C) x 20

### Probability of a score

2. Assume that the mean SAT score in Mathematics for 11th graders across the nation is 500, and that the standard deviation is 100 points. Find the probability that the mean SAT score for a randomly selected group of 150 11th graders is between 470 and 530. Please show all work for me to better understand how it was done.

### Statistics Problem Set: Discrete Probability Distribution

4.182: For each of the following examples, decide whether x is a binomial random variable and explain your decision. (see examples attached). 4.184: Consider the discrete probability distribution shown here. x 10 12 18 20 p(x) .2 .3 .1 .4 a) Calculate mean, varian

### Monte Carlo and Crystal Ball

1. Explain the difference between descriptive and prescriptive (optimization) models. 2. Describe how to use Excel data tables, scenario manager, and goal seek tools to analyze decision models. 3. Explain the purpose of Solver and what type of decision model it is used for. 4. What approaches can you use to incorporate uncert

### Calculating the Mean and Variance of a Poisson Random Variable

1. Let X be a random variable with probability density function given by f(x) - 2(1 - x), 0 <= x <= 1, 0, otherwise a. Find the density function of Y - X^2 b. Find the mean and variance of Y. 2. Let X be a random variable with probability density distribution given by f(x) - x, 0 <= x <

### Probability Exercises

1. Determine for the following box, whether number and shape are independent or dependent. The box has a triangle with a 3 in it, a square with a 2 in it, a square with a 3 in it, and a triangle with a 2 in it. a. dependent b. independent 2. There are 5 Democrats, 18 Republicans, and 14 Independents in a room. Two peopl

### Statistics Problem Set: Uniform and Exponential Distribution

86. Give the z-score for a measurement from a normal distribution for the following: a. 1 standard deviation above the mean b. 1 standard deviation below the mean c. equal to the mean d. 2.5 standard deviations below the mean e. 3 standard deviations above the mean 118. Suppose x is a binomial random vari

### Statistics Problem Set: Binomial Random Variables

38. Suppose x is a binomial random variable with n = 3 and p = 3 a. calculate the value of p(x), x = 0,1,2,3 using the formula for a binomial probability distribution b. using your answers to part a, give the probability distribution for x in tabular form 42. The binomial probability distribution is a family of proba

### Distribution and Security Analysis

2. Security analysts are professionals who devote full time efforts to evaluating the investment worth of a narrow list of stocks. The following variables are of interest to security analysts. Which are discrete and which are continuous random variables? a. the closing price of a particular stock on the New york stock exc

### Multiplicative Rule and Independent Events

48. For two events, A and B, P(A) = 0.4, P(B) = 0.2, and P(A/B)=0.6 a. find P(A and B) b. find P(B/A) 50. An experiment results in one of three mutually exclusive events, A,B, or C. It is known that P(A) =.30, P(B) = .55 and P (C) = .15. Find each of the following probabilities: a. P(A U B)

### Exponential Distribution Function and Derivation of the Equation

Let F(t) = 1- e ^(- lamda*t) a) Show how to generate a random variable from the exponential distribution function shown above. Show derivation of the equation. b) Generate 10000 random variables X and Y with cumulative distribution F(t); do this twice using lambda =2,3 (so 2 columns of 10000 numbers each). c) Using the equa

### Probability of selection from a group

We have 7 boys and 3 girls in our church choir. There is an upcoming concert in the local town hall. Unfortunately, we can only have 5 youths in this performance. This performance team of 5 has to be picked randomly from the crew of 7 boys and 3 girls. a. What is the probability that all 3 girls are picked in this team of 5?

### Jar of Marbles Probability Questions for Stats Students

Two marbles are selected, one at a time from a jar of marbles containing 10 black, 10 brown, 10 white and 10 green marbles. Let x represent the number of white marbles selected in 2 separate selections from the jar. (A) If this experiment is completed without replacing the marbles each time, explain why x is not a binomial

### Find Expected Results and Probability

1. If the IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. (a) Find the probability that a randomly selected person has an IQ score between 88 and 112. (Show work) (b) If 100 people are randomly selected, find the probability that their mean IQ score is greater than 103. (Show work) 2.

### What is a Probability Distribution?

6. Determine whether each of the distributions given below represents a probability distribution. Justify your answer. (a) x 1 2 3 4 P(x) 1/4 1/12 1/3 1/6 (b) x 3 6 8 P(x) 0.2 2/5 0.3 (c) x 20 30 40 50 P(x) 3/10 -0.1 0.5 0.3

### Finding probability and area under a distribution

1. Find the following probability for the standard normal random variables z P( -1.5< z < 1.5) 2. Find the area under the standard normal probability distribution between the following pairs of z- scores. A. Z= 0 and z= 3.00 B. z= 0 and z= 1.00 C. Z = 0 and z = 2.00 D. Z = 0 and z = 0.62 E. z = 0 and z = 0.39

### Binomial and Empirical Probability

A baseball player hits 59 home runs during a season with 161 games. What is the empirical probability that the player would hit at least one home run during any given game of the season? On the basis of the answer how many home runs would we expect the player to hit during any stretch of 8 games during the season?

### Outcomes of Variables

1. The outcomes of two variables are (Low, Medium, High) and (On, Off), respectively. An experiment is conducted in which the outcomes of each of the two variables are observed. The probabilities associated with each of the six possible outcome pairs are given in the accompanying two-way table. Low Medium High

### Probability with Dice.

Two fair dice are tossed, and the face on each die is observed. a. Use a tree diagram to find the 36 sample points contained in the same space. b. Assign probabilities to the sample points in part a. c. Find the probability of each of the following events: A = {3 showing on each die} B = {sum of two numbers showing

### Normal Distribution and Probabilities

Given a normal distribution with a mean of 100 and a standard deviation of 10, if you select a sample of n = 25, what is the probability that the sample mean is a. less than 95? b. between 95 and 97.5? c. above ?102.2? d. There is a 65% chance that the sample mean is above what value?

### Calculations of Probability

Calculate the following probabilities: a) Flipping "heads" with a normal coin five times in a row. b) Rolling a "6" with one die. c) Rolling a "7" with two dice. d) Drawing a jack OR a red card (with replacement). e) Drawing an ace, followed by a king (with replacement).

### Probability Analysis for Quality Measurements

A company that crafts home and garden features has collected some data from routine quality control studies on its mowers. The last 30 days' findings are attached as an Excel document to this post containing 200 sample weights of mower blades. They do their best to implement in-process quality checks to remain in control and man