Four white and four black balls are distributed in two urns in such a way that each contains four balls. We say that the system is in state i,i = 0,1,2,3,4 , if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth step. Explain why {Xn, n = 1, 2, 3, . . .} is a Markov chain and calculate its
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Four white and four black balls are distributed in two urns in such a way that each contains four balls. We say that the system is in state i,i = 0,1,2,3,4 , if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth step. Explain why {Xn, n = 1, 2, 3, . . .} is a Markov chain and calculate its transition matrix.
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- Suppose a certain manufacturing company produces connecting rods for 4- and 6-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2,500, and the cost required to set up the production line for the 6-cylinder connecting rods is $3,900. Manufacturing costs are $13 for each 4-cylinder connecting rod and $16 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 6,000 6-cylinder connecting rods and 8,000 4-cylinder connecting rods. Let x4 = the number of 4-cylinder connecting rods produced next week x6 = the number of 6-cylinder connecting rods produced next week s4 = 1 if the production line is set up to…Approximately one-third of all human twins are identical (one-egg) and two thirds are fraternal (two-egg) twins. Identical twins are necessarily the same sex, with female and male being equally likely. Among fraternal twins, approximately one-fourth are both female, one-fourth are both male, and half are one female and one male. Finally, among all births in theUnited States, approximately 10 in 1000 are twin births. A = {a U.S. birth results in twin females} B = {a U.S. birth results in identical twins} C = {a U.S. birth results in twins} Find P(A ∩ B) Find P(A ∩ C). Find P(B ∩ C).Approximately one-third of all human twins are identical (one-egg) and two thirds are fraternal (two-egg) twins. Identical twins are necessarily the same sex, with female and male being equally likely. Among fraternal twins, approximately one-fourth are both female, one-fourth are both male, and half are one female and one male. Finally, among all births in theUnited States, approximately 15 in 1000 are twin births. A = {a U.S. birth results in twin females} B = {a U.S. birth results in identical twins} C = {a U.S. birth results in twins} Find P(A ∩ C). Compare P(A ∩ C) with P(A)*P(C) Is the event pairwise independent?
- Approximately one-third of all human twins are identical (one-egg) and two thirds are fraternal (two-egg) twins. Identical twins are necessarily the same sex, with female and male being equally likely. Among fraternal twins, approximately one-fourth are both female, one-fourth are both male, and half are one female and one male. Finally, among all births in theUnited States, approximately 15 in 1000 are twin births. A = {a U.S. birth results in twin females} B = {a U.S. birth results in identical twins} C = {a U.S. birth results in twins} Find P(A ∩ B) Find P(A ∩ C). Find P(B ∩ C). Which event is pairwise independent?The dependencies between the 10 activities listed in the table above are as follows:• The project starts with two activities—A, and B—which can be done concurrently.• When activity A is finished, activity C can start• When activity B is finished, activities D and E can start• When both activities C and D are finished, activity F can start• When activity E is finished, activity G can start• When activity F is finished, activity H can start• When both activities G and H are finished, activity I can start• When activity G is finished, activity J can start• The project is complete when activities I and J are finished Questions1. Organize the information above in an activity precedence table showing the activity, precedent and duration (calculated using the PERT weighted average formula and round to the nearest integer).2. Draw an AON network diagram and calculate its early start, early finish, late start and late finish times, floats (Free and total) for each activity. Make sure you…Suppose in a small town named Oshiland there are three places to eat, a Chinese restaurant, a Mexican restaurant, and a pizza place. Everyone in town eats dinner in one of these places or has dinner at home. Assume that 20% of those who eat in the Chinese restaurant go to Mexican next time, 20% eat home and 30% go to the pizza place. From those who eat in the Mexican restaurant, 10% go to the pizza place, 25% go to the Chinese restaurant, and 25% eats at home next time. From those who eat at the pizza place, 30% eat at home, 30% eat at the Chinese restaurant, and 10% eat at the Mexican restaurant next time. Those who eat at home, 20% go to the Chinese restaurant, 25% go to Mexican restaurant, and 30% to the pizza place. a. Set us the matrix of transition probabilities. b. Find the steady state probabilities. c. In the long run, which restaurant will have the most customers?
- Suppose that a region has two hospitals. Hospital A has about 10 births per day, and hospital B has about 50 births per day. About 50% of all babies are boys, but the percentage who are boys varies at each hospital from day to day. Over the course of a year, which hospital will have more days on which 60% or more of the births are boys? The smaller Hospital A, the larger B, or will Hospital A and B have nearly the same number of days?How many different teams of 4 can be chosen from a group of 16 adults and 18 children if each téam must have at least two adults on it?Consider two small companies, each with 9 employees including 1 boss with the highest salary in each company, and 8 people working for them with lower salaries. The table below gives the annual salaries of the people in each company. These salaries are identical except for the fact that Small Company B is more profitable overall, and their boss decides to keep a much higher salary than the boss of Small Company A. Small Company A Small Company B $85 000 (boss A) $360 000 (boss B) $65 000 $65 000 $60 000 $60 000 $40 000 $40 000 $35 000 $35 000 $35 000 $35 000 $35 000 $35 000 $30 000 $30 000 $30 000 $30 000 For each of the two companies, calculate the mean, median, and mode, and show your answers and calculations in the space below (rounded to the nearest thousands place). After you answer this question, answer the two follow-up questions about the results you just calculated.
- 7.) Suppose that thirty-two students signed up for classes during an orientation session. If exactly twenty of them signed up for Chemistry and exactly sixteen of them signed up for English, how many of them signed up for both Chemistry and English?Suppose in a small town there are three places to eat, a Chinese restaurant, a Mexican restaurant, and a pizza place. Everyone in town eats dinner in one of these places or has dinner at home. Assume that 20% of those who eat in the Chinese restaurant go to Mexican next time, 20% eat home and 30% go to the pizza place. From those who eat in the Mexican restaurant, 10% go to the pizza place, 25% go to the Chinese restaurant, and 25% eats at home next time. From those who eat at the pizza place, 30% eat at home, 30% eat at the Chinese restaurant, and 10% eat at the Mexican restaurant next time. Those who eat at home, 20% go to the Chinese restaurant, 25% go to Mexican restaurant, and 30% to the pizza place. FIND THE STEADY STATE PROBABILITIES