Can somebody please help me with these problems. I am so confused by them. If you answer all correctly I will rate positive :)

Roulette is one of the most common games played in gambling casinos in Las Vegas and elsewhere. An American roulette wheel has slots marked with the numbers from 1 to 36 as Well as 0 and 00 (the latter is called "double zero"). Half of the slots marked 1 to 36 are colored red and the other half are black. (The 0 and 00 are colored green.) With each spin of the wheel, the ball lands in one of these 38 slots. one of the many possible roulette bets is to bet on the color of the slot that the ball will land on (red or black) If a player bets on red, he wins if the outcome is one of the 18 red outcomes, and he loses if the outcome is one of the 18 black outcomes or is 0 or 00. So, when betting on red, there are 18 outcomes in which the player wins and 20 outcomes in which the player loses Therefore, when betting on red, the probability of winning is 18/38 and the probab of losing is 20/381 When betting on red, the payout for a win is "1 to 1". This means that the player gets their original bet back PLUS and additional amount equaling their bet. In other words, they double their money. (Note: If the player loses they lose whatever amount of money they bet.) Scenario 1: Mike goes to the casino with $400. His plan is to bet $100 on red 9 consecutive times or until he either has increased his total to $500 or has lost all of his money. What is the probability he Will go bankrupt (i.e. end up with $0)? (Give your answer correct to four decimal places) (Hint: Set this problem up as an absorbing Markov Chain with 6 states where the states keep track of his current amount of money. The amount of money he has will always be $0, $100, $200, $300, $400, or $500. The states where he has $0 or $500 are absorbing states since he quits playing whenever one of these states is reached. Scenario 2: Mike goes to the casino with $400 and will still be betting on red. As in Scenario 1, he will bet $100 each time he places a bet. However, instead of limiting himself to a maximum of 9 bets he decides to play indenitely until he has reached $500 or goes bankrupt, If he uses this betting method, what is the probability he will eventually go bankrupt? (Give your answer correct to four decimal places) The victims of a certain disease being treated at Wake Medical Center are classied annually as follows: cured, in temporary remission, sick, or dead from the disease. Once a patient is Cured, he is permanently immune. Each year, those in remission get sick again with probability 0.06, are cured with probability 0.29, die with probability 0'05, and stay in remission with probability ooso Those who are sick are cured with probab ty 0'05, die with probability 0'2, go into remission with probability 0.4, and remain sick with probability 0.35. Find the transition matrix and do the calculations necessary to answer the following questions. (Give your answers correct to three decimal places.) (a) If a patient is now in remission, what is the probability he is still alive in two years? (Hint: in which states is a patient alive?) (b) If a patient is now in remission, what is the probability he dies within three years? (c) On average, how many years will a patient In remission live before bean cured or dying from the disease? years (d) If a patient is presently sick, what is the expected number of years before the patient is cured or dies? years (e) What Is the probability that someone who Is currently In remission will eventually be cured? (f) what is the probability that someone who is currently sick will eventually be cured