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This page demonstrates how experiments with random data converge toward the predictions of probability theory as more and more experiments are run.
The panel above initially shows the bell-curve normal distribution approximating the binomial distribution for an experiment consisting of 16 coin flips. To see the bell curve for experiments with different numbers of flips, enter a value between 4 and 1024 inclusive in the “Flips/run” box before pressing “Start”. For large numbers of flips, the bell curve will be very narrow, since the probability of a large excess of heads or tails is very low. In every case, the peak of the bell curve, the most probable result, represents an equal numbers of heads and tails.
Pressing the “Start” button begins a series of simulated experiments, each consisting of the number of flips specified by “Flips/run”. The number of heads are tallied and displayed as a histogram superimposed on the normal curve. As more and more experiments are run, the scale of the histogram bars is adjusted so the tallest bar remains as high as the peak in the normal curve. Press “Stop” to suspend the running of experiments; press “Run” to resume after a pause. “Reset” stops the experiment if running and clears the result for a new run. If you like, you can enter a new value for “Flips/run” whenever the experiment is stopped; previous results will be cleared. The “Step” button runs one experiment and updates the histogram each time it is pressed.
Observe how at the outset, especially for experiments with relatively few flips, results may appear to depart substantially from the chance expectation, but as more and more experiments are run, the results converge ever more closely on the prediction from probability theory. The initial outlying results “scroll down” as more and more experiments produce the most probable outcomes.