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Example 323 A coin-tossing surprise A fair coin is repeatedly ipped until the last three tosses either show the combination TTH or the combination THH Here H means that the outcome of a toss is a head and T that it is a tail What is the probability that the combination TTH occurs before the combination THH To answer this question, we de ne a Markov chain with eight states, including two absorbing states Let state 0 mean the beginning of a game, state 1 = the rst toss is H , state 2 = the rst toss is T , state 3 = the last two tosses show HH, state 4 = the last two tosses show HT, state 5 = the last two tosses show TT, state 6 = the last two tosses show TH, state 7 = the last three tosses show TTH and state 8 = the last three tosses show THH The states 7 and 8 are taken absorbing It is implicit in the de nition of the states 3, 4, 5, 6 that the combinations TTH and THH have not appeared before The Markov chain that describes the evolution of the state of the system has the one-step transition probabilities p01 = p02 = 1 , p13 = p14 = 1 , p25 = p26 = 1 , 2 2 2 p33 = p34 = 1 , p45 = p46 = 1 , p55 = p57 = 1 , 2 2 2 p63 = p68 = 1 , p77 = 1, p88 = 1, the other pij = 0 2 The Markov chain will ultimately be absorbed in one of the states 7 and 8 (this fact can formally be proved by proceeding as in the proof of Theorem 322 below and replacing the states 7 and 8 by a single absorbing state) Denote by fi the probability that the Markov chain is ultimately absorbed in state 7 starting from state i The probability f0 gives the desired probability that the combination TTH occurs before the combination THH The probabilities f0 , , f6 satisfy a system of linear equations The equation for fi follows by conditioning on the next state after the current state i This gives f0 = 1 f1 + 1 f2 , 2 2 f3 = 1 f3 + 1 f4 , 2 2 f5 = 1 f5 + 2.

f 2 = 1 f5 + 1 f6 , 2 2

CAP-GPRS-ReferenceNumber destinationReference originationReference :: = SEQUENCE { [0] Integer4 [1] Integer4

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struction of colossal stadiums at Ohio State, Michigan, Notre Dame, Northwestern, the University of Chicago (which was a charter member of the Big Ten under Amos Alonzo Stagg), and the University of Illinois, where Red Grange ran wild before signing with the Chicago Bears to play in sparkling new Soldier Field, which many believe saved the embryonic National Football League (NFL) from extinction It was an era when anything seemed possible, from Lucky Lindbergh s trans-Atlantic flight to Hollywood s first talkie (Al Jolson s Jazz Singer) to a fundamentally new form of literature direct, spartan, and scandalous for its adult themes ushered in by Oak Park, Illinois, native Ernest Hemingway Margaret Sanger introduced the diaphragm to the public, while drugstores like Walgreens began selling condoms, the only thing proven during the Great War to stop the epidemic spread of venereal disease The Twenties acted like a flask of bootleg whiskey on the American psyche, reducing inhibitions and inciting silliness, but also creating a climate that cultivated misconduct not to mention spawning the national hangover that followed The Harding administration s troubles only confirmed the public s feelings of betrayal, which peaked in 1922 when investigators discovered Harding s Secretary of the Interior had been selling drilling rights on federal lands in Teapot Dome, Wyoming, in exchange for hundreds of thousands of dollars in kickbacks, stock, and cattle Nothing encouraged widespread lawlessness, however, like the Nineteenth Amendment: Prohibition Even average, otherwise law-abiding citizens, including Charles Walgreen himself (he definitely broke this law), found it impossible to follow this edict every day Terms like moonshine, bootlegging, and organized crime all entered the American lexicon People would go to incredible lengths to get their lips on a few sips of alcohol, in just about any form they could get it The production of legal sacramental wine, for example, increased by hundreds of thousands of gallons, suggesting either an amazing upturn in the number of church-goers or an epidemic of illicit alcohol use among everyday Americans26.

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The solution of these equations is (f0 , , f6 ) = ( 2 , 2 , 2 , 2 , 2 , 1, 1 ) The desired 3 3 3 3 3 3 probability is thus 2 A surprising result for many people Can you give a simple 3 explanation why the sought probability is not equal to 1 2

322 Mean First-Passage Times Example 311 asked how to nd the expected number of steps the drunkard needs to return to his starting point More generally, consider a Markov chain {Xn } for which (a) the state space I is nite, (b) there is some state r such that for each state i I there is an integer n(= ni ) (n) such that pir > 0 What is the mean return time from state r to itself Let = min{n 1 | Xn = r}, To calculate rr = E( |X0 = r), we need the mean visit times ir = E( | X0 = i) for each state i = r By conditioning on the next state after state r, rr = 1 +

OPTIONAL, OPTIONAL}

(323)

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