74. In this version of “dice blackjack,” you toss a singledie repeatedly and add up the sum of your dice tosses.Your goal is to come as close as possible to a total of 7without going over. You may stop at any time. If yourtotal is 8 or more, you lose. If your total is 7 or less,the “house” then tosses the die repeatedly. The housestops as soon as its total is 4 or more. If the housetotals 8 or more, you win. Otherwise, the higher totalwins. If there is a tie, the house wins. Consider the following strategies:■ Keep tossing until your total is 3 or more.■ Keep tossing until your total is 4 or more.■ Keep tossing until your total is 5 or more.■ Keep tossing until your total is 6 or more.■ Keep tossing until your total is 7 or more.For example, suppose you keep tossing until yourtotal is 4 or more. Here are some examples of how thegame might go:■ You toss a 2 and then a 3 and stop for total of5. The house tosses a 3 and then a 2. You losebecause a tie goes to the house.■ You toss a 3 and then a 6. You lose■ You toss a 6 and stop. The house tosses a 3 andthen a 2. You win.■ You toss a 3 and then a 4 for total of 7. The housetosses a 3 and then a 5. You win.Note that only 4 tosses need to be generated for thehouse, but more tosses might need to be generatedfor you, depending on your strategy. Develop a simulation and run it for at least 1000 iterationsfor eachof the strategies listed previously. For each strategy,what are the two values so that you are 95% surethat your probability of winning is between thesetwo values? Which of the five strategies appears tobe best?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
74. In this version of “dice blackjack,” you toss a singledie repeatedly and add up the sum of your dice tosses.Your goal is to come as close as possible to a total of 7without going over. You may stop at any time. If yourtotal is 8 or more, you lose. If your total is 7 or less,the “house” then tosses the die repeatedly. The housestops as soon as its total is 4 or more. If the housetotals 8 or more, you win. Otherwise, the higher totalwins. If there is a tie, the house wins. Consider the following strategies:
■ Keep tossing until your total is 3 or more.
■ Keep tossing until your total is 4 or more.
■ Keep tossing until your total is 5 or more.
■ Keep tossing until your total is 6 or more.
■ Keep tossing until your total is 7 or more.For example, suppose you keep tossing until yourtotal is 4 or more. Here are some examples of how the
game might go:
■ You toss a 2 and then a 3 and stop for total of
5. The house tosses a 3 and then a 2. You losebecause a tie goes to the house.
■ You toss a 3 and then a 6. You lose■ You toss a 6 and stop. The house tosses a 3 and
then a 2. You win.
■ You toss a 3 and then a 4 for total of 7. The housetosses a 3 and then a 5. You win.Note that only 4 tosses need to be generated for the
house, but more tosses might need to be generatedfor you, depending on your strategy. Develop a simulation and run it for at least 1000 iterationsfor eachof the strategies listed previously. For each strategy,what are the two values so that you are 95% surethat your probability of winning is between thesetwo values? Which of the five strategies appears tobe best?
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