ne bets $1. 0.45, he wins the game and with probability 1 - p = 0.55, he loses the game. His goal is to increase his capital to $3, and as soon as he does, the game is over. The game is also over if his capital is reduced to zero. Construct an absorbing Markov chain and answer the following questions. • What is the expected duration of the game? • What is the probability that he goes broke?

Assume that a website www.funwithmath1600.ag has three pages: • Page A: King Algebra • Page B: Learn1600andWin Page C: Linear AlgbraIsEverywhere Each page has some links to the other pages of this website and no pages links to any page outside this website. • Page A has three links to page B and only one link to page C. • Page B has three links to page A and two links to page C. • Page C has one link to page A and two links to page B. A student decides to explore this website starting from page A. Since reading content is always a boring task (is it?!) they decide to choose one of the links in page A with equal probability and click on the link to see the next page. As a result, on the next step, they will end up on page B with probability 3/4 and on the page C with probability 1/4. This process is then continued by the student with the same rule: Go the next page by clicking, with equal probability, on one of the existing links that are on the present page. (Use only fractions in your…

2. This year Karen has decided to diversify her business by growing broc- coli as well as lettuce. Karen plants lettuce with a probability of or broccoli with a probability of . By the time these plants are ready to harvest the price of lettuce increases with a probability of and decreases with a probability of 3. The price of broccoli increases with a probability of , decreases with a probability of or stays the same with a probability of . (a) Draw a probability tree for this scenario. (b) Use the probability tree to determine the probability that Karen plants broccoli and that the price of broccoli increases by the time it is ready to harvest. (c) Use the probability tree to determine the probability that the price of whatever Karen plants increases by the time it is ready to harvest. (d) Calculate Pr(Price of lettuce does not change | Karen plants lettuce). (e) Calculate Pr(Price of broccoli does not changen Karen plants broccoli).

An airline offers "advance-purchase" fares to customers who buy their tickets at least four weeks before the flight. The airline has noticed that 80% of its customers who buy tickets don't take advantage of advance-purchase fares and the remaining 20% do. The no-show rate for those who don't have advance-purchase fares is 25%, while for those who do have them, it is only 15%. (a) Calculate the probabilities labeled on the tree diagram below. (Give answers correct to three decimal places.) A tree diagram with 6 branches is shown. The tree diagram begins at a point and an upper and lower branch extend from this point to the right. From each of these two branches two more upper and lower branches extend to the right. The branches extending from the first point from top to bottom are labeled a and b and end at points labeled "no advance purchase" and "advance purchase" respectively. The branches extending from "no advance purchase" and "advance purchase" from top to bottom are labeled…