Probability Calculation: Calculate the probability of the events happening with the given information. Please write BOTH the probability and the percentage chance.

1). You are playing a game with Dungeons and Dragons with some

friends. In order to defeat the goblins that are defending a magical

sword you have to either roll a 6 or higher on a 12 sided die OR roll

lower than 14 on a 20 sided die. What is the probability that you

defeat the goblins provided that you roll both dies? (Hint: Remember

that rolling the 12 sided die, and rolling the 20 sided die are

independent events).

2). You decide to go to Keenland with some friends. You reckon that

horse number 4 has a 7:1 chance of winning. What is the probability

that this horse wins?

If a gambler rolls two dice and gets a sum of seven, he wins $10, and if he gets a sum of four, he wins $25. The cost to play the game is $5.

A. Give the probability distribution of the winnings for this game.

B. What are the expected winnings for a person who plays this game?

Elevator (C++)

Following the diagram shown below, create the class Elevator. An Elevator represents a moveable carriage that lifts passengers between floors. As an elevator operates, its sequence of operations are to open its doors, let off passengers, accept new passengers, handle a floor request, close its doors and move to another floor where this sequence repeats over and over while there are people onboard. A sample driver for this class is shown below. Each elevator request translates into just a single line of output shown below. You should probably make a more thorough driver to test your class better.

An elevator system in a tall building consists of a 800-kg car and a 950-kg counterweight joined by a light cable of constant length that passes over a pulley of mass 280 kg. The pulley, called a sheave, is a solid cylinder of radius 0.700 m turning on a horizontal axle. The cable does not slip on the sheave. A number n of people, each of mass 80.0 kg, are riding in the elevator car, moving upward at 3.00 m/s and approaching the floor where the car should stop. As an energy-conservation measure, a computer disconnects the elevator motor at just the right moment so that the sheave–car– counterweight system then coasts freely without friction and comes to rest at the floor desired. There it is caught by a simple latch rather than by a massive brake. (a) Determine the distance d the car coasts upward as a function of n. Evaluate the distance for (b) n = 2, (c) n = 12, and (d) n = 0. (e) For what integer values of n does the expression in part (a) apply? (f) Explain your answer to part (e). (g) If an infinite number of people could fit on the elevator, what is the value of d ?

A manufacturing process produces piston rings, with ID (inner diameter) dimension as shown above.
Process variation causes the ID to be normally distributed, with a mean of 10.021 cm and a standard
deviation of 0.040 cm.
a. What percentage of piston rings will have ID exceeding 10.075 cm? What percentage of piston rings
will have ID exceeding 10.080 cm? (4)

b. What is the probability that a piston ring will have ID between 9.970 cm and 10.030 cm? (This is the
customer’s specification that the supplier tries to provide .)
(ie.) If the specification is “9.970cm < ID < 10.030cm”, what %’age of piston rings are “out of spec”? (4)

c. Half (50%) of all piston rings have ID below 10.021 cm. What is the dimension corresponding to
the smallest 10%, and what is the dimension corresponding to the largest 10%? What is the
dimension corresponding to the smallest 20%, and what is the dimension corresponding to the
largest 20%

d. Piston rings with ID too small or too large have different problems that the assembly operation
(customer) would like to know about in advance.
From your results in b., how many parts in a production run of 5000 pieces would be above
specification? How many parts would be below specification?

Parts with large ID out-of-specification are charged back to the vendor at $2.00 each. Parts with
small ID out of specification are charged back at $1.25 each. What is the total expected penalty
cost (for the vendor) for the 5000 pieces?

1. Nobel is a serial gambler and regularly plays several rounds of a gamble in which he wins $10,000 if the head comes on top twice when a fair coin is flipped twice and loses $5,000 for any other outcome from the two flips of the coin. In other words, the outcome of the gamble is determined by flipping a coin twice on each round of the gamble and Nobel wins only if head comes on top on both flips of the coin and loses if any other outcome occurs. Nobel is considering to play 10 rounds of such a gamble next week hoping that he will win back the $20,000 he lost in a similar gamble last week. If we let X represent the number of wins for Nobel out of the next 10 rounds of the gamble, X can be assumed to have a binomial probability distribution. Please answer the following questions based on the information given above.

a. Please calculate the probability of success for Nobel on each round of the gamble. Show how you arrived at your answer.

b. What is the probability that Nobel will win none of the 10 rounds of the gamble?  

c. What is the probability that Nobel will lose more than 5 of the 10 rounds of the gamble?   

d. What is the probability that Nobel will win at least 8 out of the 10 rounds of the gamble?   

e. What is the probability that Nobel will lose less than 4 of the 10 rounds of the gamble?   

f. What is the probability that Nobel will lose no more than 7 out of the 10 rounds of the gamble? g. How much money is Nobel expected to win in the 10 rounds of the gamble? How much money is he expected to lose? Given your results, do you think Nobel is playing a smart gamble? Please show how you arrived at your results and explain your final answer.

h. Calculate and interpret the standard deviation for the number of wins for Nobel in the next 10 rounds of the gamble. Show your work.

2. Vehicles arrive at a toll bridge at an average rate of 180 an hour. Only one toll booth is currently open and can process arrivals (collect tolls) at a mean rate of 22 seconds per vehicle.

A. How many vehicles should be expected at the toll bridge in a 10-minute period? Please show how you arrived at your answer.

B. Define X to be the number of vehicles arriving at the toll bridge in a 10-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that 25 vehicles will arrive at the toll bridge in a 10-minute period.

C. Please calculate the probability of at least 20 vehicles arriving at the bridge within 10-minute period.                                                                  

D. Please calculate the probability of less than 28 vehicles arriving at the bridge within 10-minute period.

E. Please calculate the probability that no vehicle will arrive at the bridge within a 10-minute period.

F. Please calculate the probability of more than 35 vehicles arriving at the bridge within a 10-minute period. Show your work.   

G. Please calculate the probability of at most 22 vehicles arriving at the bridge within a 10-minute period. Show your work.

G. Calculate and interpret the standard deviation of X.   

I. For what purpose can the kind of information you have calculated above be used? If the state department of transportation wants to reduce the average wait time for the drivers to less than 25 seconds at the toll bridge, do you think they should open another toll booth at the bridge? Please explain.             

Q6 The weight of adults in USA is normally distributed with a mean of 172 pounds and a standard deviation of 29 pounds. What is the probability that a single adult will weigh more than 190 pounds?

Q7 Along the lines of Q6 above, what is the probability that 25 randomly selected adults will have a MEAN more than 190 pounds?

Q8 Along the lines of Q6 above, an elevator has a sign that says that the maximum allowable weight is 4750 pounds. If 25 randomly selected people cram into the elevator, what is the probability that it will be over the maximum allowable weight?

Q9 The human gestation period (pregnancy period) is normally distributed with a mean of 268 days and a standard deviation of 15 days. If 25 women are randomly selected, find the probability that the sample will have a mean of less than 260 days.

Q10 Along the lines of Q9 above, a random selection of 25 woman (volunteers) are put on a special diet and the sample mean is less than 260 days. Does it appear that the diet has an effect of gestation period? What could make you more “certain”?

Accountant Ian Somnia's infamous napping at work has given rise to a challenge from colleague I. M. Tarde. At the company retreat, the two will take turns trying to stay awake during a sequence of 5-minute company training films. The probability that Somnia will fall asleep during a given film is 0.8, while the probability that Tarde does is 0.7. The first one to stay awake (as determined by a panel of alert judges) wins a $1000 bonus. If Somnia and Tarde alternate watching films, with Somnia going first, what are the chances that Tarde wins the bonus? (Hint: A typical sequence of films for which Tarde wins the bonus is NNNY, where N = “not awake” and Y = “awake.”)

A local elevator moves upward at a constant 3.3 mps passing a stopped express elevator. Precisely 2.8 seconds later the express elevator starts upward with a constant acceleration and catches up with the local elevator where the velocity of the local with respect to the express elevator is -9.3 m/s. Determine (a) the acceleration of the express elevator in m/s/s and (b) the distance traveled in m for the express to catch up with the local elevator.

A) An 80- kg man stands in an elevator. What force does he exert on the floor of the elevator under the following conditions? The elevator is stationary.

B) The elevator accelerates upward at 2.2 m/s2.

C) The elevator rises with constant velocity of 4.5 m/s.

D) While going up, the elevator accelerates downward at 1.8 m/s2. The elevator goes down with constant velocity of 7.2 m/s.