1. Abigail spends her entire weekly allowance on either candy or toys. If she buys candyone week, she is 60% sure to buy toys the next week. The probability that she buystoys in two successive weeks is 25%.(a) Create a transition diagram that describes this scenario.(b) Create a transition matrix that describes this scenario. Is this scenario ergodic orabsorbing? Explain.(c) Suppose that this week, Abigail spends all of her money (100%) on candy. Usingmatrix multiplication, predict the probability that Abigail buys toys or candynext week, the following week, and three weeks from now.(d) Find the eigenvalues and eigenvectors for this transition matrix.(e) In the long run, what is the probability that Abigail will spend her money ontoys? Explain.

As per the guidelines of Bartleby once max three subparts can be answered, please repost other subparts mentioning the required subparts number. Given Abigail spends her entire weekly allowance on either candy or toys. If she buys candy one week then she buys toys next week 0.6 times and buys candy next week 0.4 times. Probability that she buys toys two consecutive weeks is 0.25. (a) To create a transition diagram to describe the scenario. (b) To create a transition matrix that describe the scenario. To determine the scenario ergodic or absorbing. (c) If one week Abigail spends all his money on candy then to find the probability of buying candy and toys in three successive weeks from now.(a) Given Abigail spends her entire weekly allowance on either candy or toys. If she buys candy one week then she buys toys next week 0.6 times and buys candy next week 0.4 times. Probability that she buys toys two consecutive weeks is 0.25. Suppose the probability of buying toy next week if she buys toy this week is b, then probability of buying toy two consecutive weeks is b2=0.25⇒b=0.5. Then the probability that she buys candy next week if she buys toy this week is 1-0.5=0.5. So the transition diagram that describe the scenario is given by (b) Suppose xn, yn are the probabilities that Abigail buys candy in nth week and toys in nth week. Then xk+1=0.4xk+0.5ykyk+1=0.6xk+0.5yk So this transition equations can be written in matrix form as Xk+1=xk+1yk+1=0.40.50.60.5xkyk=AXk Here the transition matrix is given by A=0.40.50.60.5 A Markov chain is called an ergodic chain if it is possible to go from every state to every state in finite steps.A Markov chain is called an absorbing chain if its transition matrix has at least one column where all the elements is zero except one element which is equal to 1. In this case the transition from any state to any other state is possible in finite steps, and the matrix is also regular, so the Markov chain is ergodic. (c) Suppose for a week X=10, i.e. one week she buys candy only with all her money. To evaluate the probabilities that of she buy candy or toy next week, evaluate AX. AX=0.40.50.60.510=0.40.6 Thus the probability she buy candy next week is 0.4 and toy next week is 0.6. To evaluate the probabilities that of she buy candy or toy two weeks from now, evaluate A2X. A2X=AAX=0.40.50.60.5AX=0.40.50.60.50.40.6=0.16+0.30.24+0.3=0.460.54 Thus the probability she buy candy two weeks from now is 0.46 and toy two weeks from now is 0.54. To evaluate the probabilities that of she buy candy or toy three weeks from now, evaluate A3X. A3X=AA2X=0.40.50.60.5A2X=0.40.50.60.50.460.54=0.184+0.270.276+0.27=0.4540.546 Thus the probability she buy candy three weeks from now is 0.454 and toy three weeks from now is 0.546.