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three times his stake with respective probabilities 025 and 015 His strategy is to bet $5 each time his payroll is more than $50 dollars and $10 otherwise De ne an appropriate Markov chain to compute the probability that Joe reaches his goal Also calculate the expected number of bets placed by Joe until he has gone broke or reached his goal 39 A training program consists of three parts, each having a length of one month Fifty percent of the starting students immediately pass the rst part after one month, 30% drop out before the end of the rst month and 20% take the rst part again Seventy percent of the last group pass the rst part after a second trial and the other 30% still drop out Eighty percent of the students taking the second part pass this second part after the rst trial, 10% drop out after the rst trial and the other 10% move on after a second trial of the rst part Any student streaming into the third part of the training program will complete it successfully Calculate the probability that a starting student will be successful 310 Consider a nite-state Markov chain {Xn } with no two disjoint closed sets The matrix of one-step transition probabilities is called doubly stochastic when for each column the sum of the column elements equals 1 Verify that the equilibrium distribution of such a Markov chain is a uniform distribution 311 A gambling device is tuned such that a player who wins (loses) on a given play will win on the next play with probability 025 (050) The player pays $1 for each play and receives $250 for each play that is won Use Markov chain analysis to nd out whether the game is fair or not 312 A factory has a storage tank with a capacity of 4 m3 for temporarily storing waste produced by the factory Each week the factory produces 0, 1, 2 or 3 m3 waste with respective probabilities p0 = 1 , p1 = 1 , p2 = 1 , and p3 = 1 If the amount of waste 8 2 4 8 produced in one week exceeds the remaining capacity of the tank, the excess is specially removed at a cost of $30 per cubic metre At the end of each week there is a regular opportunity to remove waste from the storage tank at a xed cost of $25 and a variable cost of $5 per cubic metre The following policy is used If at the end of the week the storage tank contains more than 2 m3 of waste, the tank is emptied; otherwise no waste is removed Use Markov chain analysis to nd the long-run average cost per week 313 In a series of repeated plays, you can choose each time between games A and B During each play you win $1 or you lose $1 You are also allowed to play when your capital is not positive (a negative capital corresponds to a debt) In game A there is a single coin This coin lands heads with probability 1 ( = 0005) and tails with probability 2 1 + In game B there are two coins One coin lands heads with probability 1 and 2 10 the other coin lands heads with probability 3 If you play game B, then you must take 4 the rst coin when your current capital is a multiple of 3 and you must take the other coin otherwise In each play of either game you win $1 if the coin lands heads and you lose $1 otherwise (a) Use Markov chain analysis to verify that the long-run fraction of plays you win is 04957 when you always play game B (Hint : a three-state Markov chain suf ces) (b) Suppose you alternately play the games A, A, B, B, A, A, B, B, Use an appropriate Markov chain to verify that the long-run fraction of plays you win is 05064 This problem shows that in special cases with dependencies, a combination of two unfavourable games may result in a favourable game This paradox is called Parrondo s paradox after the Spanish physicist Juan Parrondo 314 At the beginning of each day, a crucial piece of electronic equipment is inspected and then classi ed as being in one of the working conditions i = 1, , N Here the working condition i is better than the working condition i + 1 If the working condition is i = N the piece must be replaced by a new one and such an enforced replacement takes two days If the working condition is i with i < N there is a choice between preventively replacing.

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The next example of a discrete probability distribution is called the binomial distribution One of the most commonly occurring random variables is the one that takes one of two values each time the experiment is performed Examples of this include tossing a coin, the result of which is a head or a tail; a newborn child is a female or male; a vaccination against the u is successful or nonsuccessful Examples of this situation are very common We call these experiments binomial since, at each trial, the result is one of two outcomes, which, for convenience, are called success (S) or failure (F ) We will make two further assumptions: that the trials are independent and that the probabilities of success or failure remain constant from trial to trial In fact, we let P(S) = p and P(F ) = q = 1 p for each trial It is common to let the random variable X denote the number of successes in n independent trials of the experiment Let us consider a particular example Suppose we inspect an item as it is coming off a production line The item is good (G) or defective (D) If we inspect ve items, the sample space then consists of all the possible sequences of ve items, each G or D The sample space then contains 25 = 32 sample points We also suppose as above that P(G) = p and P(D) = q = 1 p, and if we let X denote the number of good items, then we see that the possible values of X are 0, 1, 2, 3, 4, or 5 Now we must calculate the probabilities f each of these events If X = 0, then, unfortunately, none of the items are good so, using the independence of the events and the associated sample point, P(X = 0) = P(DDDDD) = P(D) P(D) P(D) P(D) P(D) = q5 How can X = 1 Then we must have exactly one good item and four defective items But that event can occur in ve different ways since the good item can occur at any one of the ve trials So P(X = 1) = P(GDDDD or DGDDD or DDGDD or DDDGD or DDDDG = P(GDDDD) + P(DGDDD) + P(DDGDD) + P(DDDGD) + P(DDDDG) = pq4 + pq4 + pq4 + pq4 + pq4 = 5pq4 Now P(X = 2) is somewhat more complicated since two good items and three defective items can occur in a number of ways Any particular order will have probability q3 p2 since the trials of the experiment are independent We also note that the number of orders in which there are exactly two good items 5 must be 2 or the number of ways in which we can select two positions for the.

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