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Unformatted text preview: MAT 183 Practice Test #3, Fall 2008; Ver. P1 YOUR NAME Problem 1. For a certain group of states, it was observed that 80% of the Democratic governors were succeeded by Democrats and 20% by Republicans. Also, 40% of the Republican governors were succeeded by Democrats and 60% by Republicans. (i) Set up the 2 × 2 stochastic matrix describing this Markov process. (ii) In the long run, what proportion of governors will be Republicans? Problem 2. With respect to a certain gene, geneticists classify individuals as dominant, recessive, or hybrid. In an experiment, individuals are crossed with hybrids, then their offspring are crossed with hybrids, and so on. For dominant individuals, 75% of their offspring will be dominant and 25% will be a hybrid. For the recessive individuals 50% of their offspring will be recessive and 50% hybrid. For hybrid individuals (to be crossed with hybrids) their offspring will be 25% dominant, 25% recessive, and 50% hybrid. In the long run what percent of the individuals in a generation will be dominant? Problem 3. A math clinic has 82 graphing calculators to lend. Each day there is a 2% chance that a given calculator will break and an 80% chance that a given broken calculator will be repaired. In the long run, about how many calculators will be broken? Problem 4. Consider a game of chance with the following characteristics: A person repeatedly bets $1 each play. If he wins, he receives $2 (in addition to his $ 1 bet which he get back). If he goes broke, he stops playing. Also, if he accumulates at least $4 , he stops playing. On each play the probability of winning is . 3 and of losing . 7 . (i) If the gambler begins with $2 , what is the expected number of times that he will play before quitting? (ii) What is the probability of losing the game if the player begins with $3 ? (iii) What is the probability of winning the game if the player begins with $1 ? Problem 5. The managers in a company are classified as top managers, middle managers, and first-line managers. Each year, 10% of top managers retire, 10% leave the company, 60% remain top managers, and 20% are demoted to middle managers. Each year,demoted to middle managers....
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