ABSTRACT
M onte Carlo simulation is a powerful probabilistic anal-ysis tool, widely used in both engineering fields and non-engineering fields. It is named after the famous gambling hot spot, Monte Carlo, in the Principality of Monaco. Chance and random outcomes are central to the modeling technique, much as they are to games like roulette, dice, and slot machines. Monte Carlo simulation was initially used to solve neutron diffusion problems in atomic bomb research at Los Alamos National Laboratory in 1944. From the time of its introduction during World War II, Monte Carlo simulation has remained one of the most-utilized mathematical tools in scientific practice. And in addition to that, it has also functioned as a very useful tool for adding an extra dimension to the teaching and learning of probability. It may help students gain a better understanding of probabilistic ideas and to overcome common misconceptions about the nature of ‘randomness’. Using computer simulation, a concrete probabilistic situation can be imitated on the computer. A key concept such as the law of large numbers can be made to come alive when students can observe the results of many simulation trials. The nature of this law is best illustrated through the coin-toss experiment. The law of large numbers says that the percentage of tosses to come out heads will be as close to 50% as you can imagine, provided that the number of coin tosses is large enough. But how large is large enough? Experiments have shown that the relative frequency of heads may continue to deviate significantly from 0.5 after many tosses, though it tends to get closer and closer to 0.5 as the number of tosses gets larger and larger. The convergence to the value 0.5 typically occurs in a rather erratic way. The course of a game of chance, although eventually converging in an average sense, is a whimsical process. To illustrate this, a simulation run of 100,000 coin tosses was made. Table 8.1 summarizes the results of this particular simulation study; any other simulation experiment will produce different numbers. The statistic Hn − 12n gives the observed number of heads minus the expected number after n tosses and the statistic fn gives the observed relative frequency of heads after n tosses. It is worthwhile to take a close look at the results in the table. You see that the realization of the relative frequency, fn, indeed approaches the true value of the probability in a rather irregular manner and converges more slowly than most of us would expect intuitively. The law of large numbers does not imply that the absolute difference between the numbers of heads and tails should oscillate close to zero. It is even typical for the coin-toss experiment that the absolute
difference between the numbers of heads and tails has a tendency to become larger and larger and to grow proportionally with the square root of the number of tosses. The mathematical explanation of this phenomenon is that the number of heads minus the number of tails after n tosses is symmetrically distributed around 0 with standard deviation
√ n (the square-root law). Figure 8.1 displays
a simulated path of the actual number of heads minus the actual number of tails for 2,000 tosses of the coin. This process is called a random walk (or drunkard’s walk), based on the analogy of an object that moves one step higher if heads is thrown and one step lower, otherwise. Results such as those shown in Figure 8.1 are not exceptional. On the contrary, in the coin-toss experiment, it is typical to find that, as the number of tosses increases, the fluctuations in the random walk become larger and larger and a return to the zero-level becomes less and less likely. This result is otherwise not in conflict with the law of large numbers, since
√ n/n goes to
zero if n gets large.
