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Probability : Exponential Density Function

#26 Please see the attached file for full problem description. a) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can model the probbility of failure of these bulbs by an exponential density function with mean mu = 1000. Find the probability that both of the bulbs fail in 1000 hours.


An octahedron is a three dimensional shape with eight sides that are equilateral triangles. THis shape is used as a die in games such as Dungeons and Dragons because all eight sides come up with equal probability. A. Assuming the sides are numbered 1 through 8, and a person throws two octahedral dice, what are the possible su

Policy Iteration : Probability Distribution and Maximizing Profit

Please see the attached file for the fully formatted problems. 1. A machine in excellent condition earns $100 profit per week, a machine in good condition earns $70 per week, and a machine in poor condition earns $20 per week. At the beginning of any week a machine can be sent out for repairs at a cost of $90. A machine s

Multivariate Probability Distributions

Please see the attached file for the fully formatted problems. Y1 & Y2 denotes the proportions of time that employee I and II actually spent working on their assigned tasks during a workday. The joint density of Y1 & Y2 is given by: f(y1,y2) = y1 + y2 , 0=<y1=<1 , 0=<y2=<1 f(y1,y2) = 0, elsewhere Employee I h


Carol and David decide to play a game as follows: Carol draws and keeps a card from a shuffled pack number 1 to 6. David the draws a card from the remaining 5. The winner is the one holding the card with the highest number. a) Determine whether or not there is an advantage to drawing the first. If the rules are no


FULL WORKINGS PLEASE. Clair and Helen frequently play each other in a series of games of table tennis. Records of the outcomes of these games show that whenever they play a series of games, Clair has a probability 0.6 of winning the first game and that in every subsequent game in the series, Clair's probability of winning the


Show that the probability that exactly one of the events A and B occurs is (see attached).

Probability Based on Random Selection

There are three boxes, each with two drawers. Box I has a gold coin in each drawer Box II has a silver coin in each drawer Box III has a gold coin in one drawer and a silver coin in the other. One box is chosen at random and a drawer is opened from that box. If it contains a gold coin, find the probability that it is in

Probability Based on Order and Random Selection

Four couples, each consisting of one man and one woman, are seated at a circular table. Assuming that each different order is equally likely, find the probability that: a) Andrew is sitting next to his partner b) Benjamin, Charles and David are sitting together (in any order) c) The men and women sit alternately


Please could I have the answer to this: Full workings please. A shortlist of 10 people is drawn up from a large number of applicants for a certain job. The shortlist consists of 7 men and 3 women. Because all the shortlisted applicants are considered to be equally qualified, the names of two of them are drawnn, one afte

Using probability with standard deviation

A particular type of electronic component for use in PCs is mass produced and subject to quality control checks since it is known that 2% of all components produced in this way are defective. The quality of a day's output is monitored as follows. A sample of 10 components is drawn from the day's output (which may be assumed to

Probabilities : Darts on a Dartboard

My uncle plays darts on a circular dart board of radius 20 cm. He assumes a dart lands anywhere on the board with equal probability. a) What is the probability that his dart lands less than 5 cm from the centre of the board? b) That his dart lands exactly 5 cm from the centre? c) My uncle wants to divide his board into ten

Probability : Bracket (Cup) System

Q. The 'cup' system for determining the champion amongst 2^n players consists of drawing lots to arrange the players in 2^n-1 pairs who are to play each other, then repeating this with the 2^n-1 winners of these matches, and so on. The winner and loser of the final match recieve the first and second prizes repectively. Suppose

An Absorbing States Problem

A mouse, after being placed in one of 4 rooms, will search for cheese in that room. If unsuccessful, after one minute it will exit to another room by selecting a door at random. (All the rooms connect to each other.) A mouse entering the room with the cheese will remain in that room. If the mouse begins in room 3, what is the pr

Probability: Birthdays on the Same Day

Determine the number of people needed to ensure that the probability at least two of them have the same day of the year as their birthday is at least 70 percent, at least 80 percent, at 90 percent, 95 percent, at least 98 percent, and at least 99 percent.

Probability: Random Selection

Two balls are drawn in succession from a box containing 4 red and 2 white balls. a) What is the probability of drawing a red ball on the second draw if the first one is not put back in the box after it is drawn? b)What is the probability of drawing a red ball on the second draw if the first one is put back in the box after it


In order to test a new car, an automobile manufacturer wants to select 4 employees to test drive the car for one year. If 12 management and 8 union employees volunteer to be test drivers and the selection is made at random, what is the probability that at least one union employee is selected?

Probability Question : Expected Value

Tom and Jim decided to play the following game for points. A single die is rolled. If it shows a non-prime number, Tom receives points equal to two times the number of dots showing. If it shows a prime number, Tom loses points equal to three times the number of dots showing. What is the expected value of the game?


Each of the numbers 1 through 10 inclusive has been written on a separate piece of paper. The 10 pieces of paper have been placed in a hat. If one piece of paper is selected at random, with replacement, find the probability that the number selected is: a. greater than 3 b. even c. odd or greater than 3 d. odd or less than


In her wallet, Susan has 12 bills. 6 are $1 bills, 2 are $5 bills, 3 are $10 bills, and 1 is a $20 bill. She passes a volunteer seeking donations for the American Red Cross and decides to select 1 bill at random. Determine: a. probability she selects $ 5 bill (my ans: 1/5) b. probability she does not select a $5 bill (my ans

Probability : Counting Principle

A social security number has 9 digits. How many different S.S numbers are possible if: a. repetition of digits is permitted b. repetition of digits is not permitted c. the first digit cannot be a 0 and repetition is not permitted I need the layout (counting principle) ex. 9 8 7 - 6 5 - 2 1 4 3 - - - - - - - -

Probability : Combinations

Mr james just won 6 tickets for each of 2 consecutive Giants home football games. For the first game, Mrs James will not be able to attend so he has 5 extra tickets. He will invite 5 of his 9 closest friends from work to go with him. Mr. and Mrs. James will both attend the second game. They have 4 extra tickets and are consideri

Probability: Joint Probability Mass Function, Covariance and Variance

Let X and Y have joint probability mass function Pr{X = i, Y = j}= c(i + 1)(j + 2) for i >= 0, j >= 0, and i + j < 4. Determine a) the marginal probability mass function of X b) the probability mass function of Y c) the conditional probability mass function of X given Y = 0 d) the probability mass function of Z = X + Y

Probability: Variance, Mean and Standard Deviation

1) Suppose we have an aisle with storage racks on both sides of the aisle. The aisle is 100 feet long. A worker is stationed at one end of the aisle. The worker needs to retrieve an item from storage. Assume that the items are divided into two groups: high turnover and low turnover. The high turnover items are stored in the loc

Random Variables : Continuous R.V., Exponenetial, R.V, Mean and Variance

3) Let X be a continuous random variable with probability density function f(s)= c(1 + s^2) for -2 <= s <= 2. a) Determine c b) Determine Pr {X <= 0} c) Determine the mean of X d) Why is the previous answer fairly obvious? e) Determine the variance of X f) Compute Pr {X = 2 | X = 0} g) Determine the cumulative distribut

Probability: Moment Generating Functions and Poisson Process

1.) Let X be a discrete random variable with probability mass function Pr {X=k} = c(1+ k^2) for k= -2, -1, 0, 1, 2. a) Determine c. b) Determine Pr {X <= 0} c) Determine the mean of X d) Why is the previous answer fairly obvious? e) Determine the variance of X f) Compute Pr {X=2 | X >= 0} g) Determine the moment genera

Estimation : Binomial Distribution

Suppose T is a random variable such that P(T=k) = (k-1)C(r-1) * p^r * (1-p)^(k-r) (It is a negative binomial distribution.). I am trying to find the expected value E(r/T) (which is equal to r * E(1/T)) By (k-1)C(r-1) I mean (k-1)!/[(r-1)!*(k-r)!].