200 CHAPTER 4 Probability Simulations Calculating probabilities are sometimes painfully difficult, but simulations provide us with a very practical alternative to calculations based on formal rules. A simulation of a procedure is a process that behaves the same way as the procedure so that similar results are produced. Instead of calculating the probability of getting exactly 5 boys in 10 births, you could repeatedly toss 10 coins and count the number of times that exactly 5 heads (or simulated “boys”) occur. Better yet, you could do the simulation with a random number generator on a computer or calculator to randomly generate 1s (or simulated “boys”) and 0s (or simulated “girls”). Let’s consider this probability exercise: Find the probability that among 40 randomly selected people, at least 3 have the same birthday. For the above problem, a simulation begins by representing birthdays by integers from 1 through 365, where 1 represents a birthday of January 1, and 2 represents January 2, and so on. We can simulate 40 birthdays by using a calculator or computer to generate 40 random numbers (with repetition allowed) between 1 and 365. Those numbers can then be sorted, so it becomes easy to examine the list to determine whether any 3 of the simulated birth dates are the same. (After sorting, equal numbers are adjacent.) We can repeat the process as many times as we wish, until we are satisfied that we have a good estimate of the probability. Use technology to simulate 20 different groups of 40 birthdays. Use the results to estimate the probability that among 40 randomly selected people, at least 3 have the same birthday. Summary of Simulation Functions (see Tech Center at the end of Section 4-5): Statdisk: Select Data from the top menu, select Uniform Generator from the dropdown menu. Excel: Click Insert Function fx, select Math & Trig, select RANDBETWEEN. Copy to additional cells. TI-83, 84 Plus: Press L, select PROB from the top menu, select randInt from the menu. StatCrunch: Select Data from the top menu, select Simulate from the dropdown menu, select Uniform or Discrete Uniform from the submenu. Minitab: Select Calc from the top menu, select Random Data from the dropdown menu, select Integer from the submenu. Technology Project a. Use Data Set 6 “Births” from Appendix B to estimate the following probabilities. P(male with a birth weight greater than 3200 grams) P(female with a birth weight greater than 3200 grams) b. Repeat part (a) using Data Set 45 “Births in New York” from Appendix B, which contains records from 465,506 births. c. Compare the results from the relatively small sample in part (a) to the results from the large sample in part (b). Does the larger data set have much of an effect on the results? Big (or Very Large) Data Project
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