AMS 5 Practice 15 532017 (8:009:05) Chapter 16, 1, 2, 3, 4, 6, 7, 9 1. A box contains 10,000 tickets: 4,000 [0]s and 6,000 [1]s. And 10,000 draws will be made at random with replacement from this box. Which of the following best describes the situation, and why? (i) The number of 1s will be 6,000 exactly (ii) The number of 1s is very likely to equal 6,000, but there is also some small chance that it will not be equal to 6,000. (iii) The number of 1s is likely to be different from 6,000, but the difference is likely to be small compared to 10,000. 1) (iii), since it is rare that you will get the exact expected value, yet the deviation is also likely to be small relative to the overall sample size. 2. Repeat exercise 1 for 10,000 draws made at random without replacement from the box. 2) (i), since there are exactly 6,000 [1]s present in the box and you draw them all. 3. A gambler loses ten times running at roulette. He decides to continue playing because he is due for a win, by the law of averages. A bystander advises him to quit, on the grounds that his luck is cold. Who is right? Or are both of them wrong? 3) Both are wrong; the second if only because luck (or probability) cannot run cold; it is always there, and the first because he has the same chance of winning this time than he did the last ten times, which is an astronomically low chance. 4. (a) A die will be rolled some number of times, and you win 1 if it shows an ace [*] more than 20 of the time. Which is better: 60 rolls, or 600 rolls? Explain.

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