Simulated Coin Tossing Experiments and the Law of Large Numbers
Prepared by Ian McLeod
Perhaps the simplest way to illustrate the law of large numbers is with coin flipping experiments. If a fair coin (one with probability of heads equal to 1/2) is flipped a large number of times, the proportion of heads will tend to get closer to 1/2 as the number of tosses increases. This Demonstration simulates 1000 coin tosses. Increasing the repetitions, you can compare the paths taken in repeated experiments. Also, varying the random seed shows the results for further experiments. The law of large numbers works as well for a biased coin. In this case, the proportion of heads converges to the probability of getting heads that is indicated by the solid gray horizontal line.

To view a large number of paths, set the number of repetitions to 20 and then click the play button for the random seed.

Download the CDF file to view the simulation using the free Wolfram CDF player.

This Demonstration was inspired by Fig. 10.1 in [1] but it provides a more dynamic visualization.

[1] D. S. Moore, The Basic Practice of Statistics, New York: W. H. Freeman and Company, 2010.