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The probabilities of heads and tails in tossing a coin are very close to 1-in-2. In principle, these probabilities come from data on many coin tosses. Joe’s personal probabilities for the winner of next year’s Super Bowl come from Joe’s own individual judgment. What about the probability that we get a run of three straight heads somewhere in 10 tosses of a coin? We can find this probability by calculation from a model that describes tossing coins. That is, once we have used data to give a probability model for the random fall of coins, we don’t have to go back to the beginning every time we want the probability of a new event.

The big advantage of probability models is that they allow us to calculate the probabilities of complicated events starting from an assignment of probabilities to simple events like “heads on one toss.” This is true whether the model reflects probabilities from data or personal probabilities. Unfortunately, the math needed to do probability calculations is often tough. Technology rides to the rescue: once we have a probability model, we can use a computer to simulate many repetitions. This is easier than math and much faster than actually running many repetitions in the real world. You might compare finding probabilities by simulation to practicing flying in a computer-controlled flight simulator. Both kinds of simulation are in wide use. Both have similar drawbacks: they are only as good as the model you start with. Flight simulators use a software model of how an airplane reacts. Simulations of probabilities use a probability model. We set the model in motion by using our old friends the random digits from Table A.

We look at simulation partly because it is how scientists and engineers really do find probabilities in complex situations. Simulations are used to develop strategies for reducing waiting times in lines to speak to a teller at banks, in lines to check in at airports, and in lines to vote during elections. Simulations are used to study the effects of changes in greenhouse gases on the climate. Simulations are used to study the effects of catastrophic events, such as the failure of a nuclear power plant, the effects on a structure of the explosion of a nuclear device, or the progression of a deadly, infectious disease in a densely populated city.

We also look at simulation because simulation forces us to think clearly about probability models. We’ll do the hard part (setting up the model) and leave the easy part (telling a computer to do 10,000 repetitions) to those who really need the right probability at the end.