STATISTICS IN SUMMARY
Chapter Specifics
• We can use random digits to simulate random outcomes if we know the probabilities of the outcomes. Use the fact that each random digit has probability 0.1 of taking any one of the 10 possible digits and that all digits in the random number table are independent of each other.
• To simulate more complicated random phenomena, string together simulations of each stage. A common situation is several independent trials with the same possible outcomes and probabilities on each trial. Other simulations may require varying numbers of trials or different probabilities at each stage or may have stages that are not independent so that the probabilities at some stage depend on the outcome of earlier stages.
• The key to successful simulation is thinking carefully about the probability model. A tree diagram can be helpful by giving the probability model in graphical form.
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In Chapter 18, we discussed probability models and the basic rules of probability. These allow us to compute the probabilities of simple events, but the math needed to find the probabilities of complicated events is often tough. In this chapter, we learn how we can use simulations to determine the probabilities of complicated events. The underlying idea, introduced in Chapter 17, is that probability is the long-run proportion of times an event occurs. Simulating such an event many, many times using technology allows us to estimate this long-run proportion.
CASE STUDY EVALUATED Look again at the Case Study that opened the chapter. For simplicity, assume that there are always nine horses in a race so that the probability of receiving one of the three inside positions is 1-in-3 if assignments are random. Use what you have learned in this chapter to describe how you would do a simulation to estimate the probability that, somewhere in a sequence of 1000 races, a string of 35 consecutive races would occur in which at least 30 times one of the three inside positions was assigned. Begin by describing how you would simulate one case of 1000 races and, for these races, how you would determine whether somewhere in the 1000 races there was a string of 35 consecutive races in which at least 30 times one of the three inside positions was assigned. Do not try to carry out the simulation. This would be very time-consuming and is best left to a computer. (In fact, the Ohio Racing Commission hired a statistician to compute the probability that in a sequence of 1000 races with varying numbers of horses, there would occur a string of 35 consecutive races in which one would receive one of the three inside positions at least 30 times. The statistician used simulation to estimate this probability.)
Online Resources
• The StatBoards video Simulation Basics discusses the basics of designing and running a simulation in the context of an example.