190 CHAPTER 4 Probability 36. Design Your Own Lottery Repeat the preceding exercise for a lottery with 6 numbers selected from 1 to 50. a. What is the probability of winning with one ticket? b. What should be the winning prize if you want to average a profit of 50%, which is common for lotteries? c. Would this lottery have much appeal? 37. Computer Variable Names A common computer programming rule was that names of variables must be between one and eight characters long. The first character can be any of the 26 letters, while successive characters can be any of the 26 letters or any of the 10 digits. For example, allowable variable names include A, BBB, and M3477K. How many different variable names are possible? (Ignore the difference between uppercase and lowercase letters.) 38.High Fives a. Ten “mathletes” celebrate after solving a particularly challenging problem during competition. If each mathlete high fives each other mathlete exactly once, what is the total number of high fives? b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can ten mathletes be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can n mathletes be seated at a round table? 39.Pick 10 Lottery For the New York Pick 10 lottery, the player first selects 10 numbers from 1 to 80. Then there is an official drawing of 20 numbers from 1 to 80. The prize of $500,000 is won if the 10 numbers selected by the player are all included in the 20 numbers that are drawn. Find the probability of winning that prize. 40. Stirling’s Approximation Stirling’s approximation given below can be used to approximate values of factorials. Use it to approximate the number of different ways that 60 ticket holders can stand in a line. How does the result compare to the exact value of 8.320987113 * 1081? Stirling’s approximation: n! ≈ 22pna n eb n where e = 2.718281828459c 4-4 Beyond the Basics Key Concept In this section we use simulations as one approach to determining when sample results are significantly low or high, so that claims about population parameters can be tested. Simulations can also be used for solving many probability problems that cannot be easily solved using the methods from the preceding sections of this chapter. We begin by defining a simulation. 4-5 Simulations for Hypothesis Tests DEFINITION A simulation of a procedure is a process that behaves the same way as the procedure, so that similar results are produced.
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