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hw2 - contradiction became known as the Paradox of the...

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Statistics 120A Assignment 2 Due Thursday, October 11 at lecture. Reading Chapter 2.1.-2.3.6 and 3.1-3.3 Exercises 1.7.5, 1.7.8, 1.8.6, 1.8.18, 1.9.8, 2.1.9, 2.1.10, 2.2.8, 2.2.16, 2.3.2, 2.3.8 Simulation The Chevalier De Mere, a French nobleman of the 17th Century, like to bet on the dice. He used to bet on the event that with 4 rolls of a die, at least one ace would turn up; the ace is the “one spot.” He also used to bet on the event that with 24 rolls of a pair of dice, at least one double-ace would turn up; the double ace is a pair of “one spots.” He reasoned thus: In one roll of a die, I have 1/6 chance to get an ace. So in 4 rolls, I have 4/6 or 2/3 chance to get at least one ace. In one roll of a pair of dice, I have a 1/36 chance to get a double ace. So in 24 rolls, I have 24/36 or 2/3 chance to get at least one ace. The chances of winning such a bet, by this reasoning, were thus equal. But experience was proving otherwise, and the first event appeared a bit more likely than the second.
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Unformatted text preview: contradiction became known as the Paradox of the Chevalier de Mere. (1) Visit the Probability link under simulation on the class website. Select Chapter 1, where you will find links to DeMere1 and DeMere2. DeMere1 simulates the first bet, DeMere2 the second. Use these to find, by simulation, the chance of winning the bets; recall that 100 replications has about a 5% margin of error, while 10,000 replications has about a 1% margin of error. (2) Notice that De Mere has reasoned in a way that suggests certain events are exclusive, so he added the chance of an ace 4 times and the chance of a double ace 24 times. Is he correct? (3) Find the correct probabilities. Hint: In 4 rolls of a die, for example, the chance of at one ace is 1 minus the chance of no aces. (4) If you have made it this far, and you have general solutions to the problem, then show that your calculations confirm the results of your simulations. 1...
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