Industrial and Management Systems Engineering



Law of Large Numbers: Comparing Relative versus Absolute Frequency of Coin Flips

Date of this Version



P. Savory (2009), “Law of Large Numbers - Coin Flip Example Comparing Relative versus Absolute Frequency”, Mathematica Software Demonstration, The Wolfram Demonstations Project. Available at:


This Mathematica demonstration showcases the law of large numbers, a key theorem in probability theory, that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. For this Demonstration, think about the simple case of tossing a fair coin. If each flip is independent of the next (so that the result of one flip does not change the probabilities of seeing heads or tails on the next flip), then the proportion of heads in tosses should get close to 0.5 as gets large.

If you were to flip a coin 10,000 times, you would expect the number of heads to be approximately equal to the number of tails when using a fair coin. The absolute difference plot can show quite large differences in absolute terms, , as the number of tosses increases. In comparison, the relative difference plot shows that in relative terms, , the difference converges to zero.