# The Law of Large Numbers

The Law of Large Numbers The Law of Large Numbers states that as the number of trials of a random process increases, the percentage difference between the expected and actual values converges to zero. In other words, the percentage error between the theoretical and experimental will closes to zero as more trials are performed. The Law of Large Numbers makes easier to guess approximate number of outcome of any event for certain number of trials such as tossing a coin n number of times.

A toss of coin has two possible events, head or tail. When we toss a coin, we never know exactly what will be outcome, head or tail as the probability of both outcome are 0. 5 or 50%. In most cases, if we toss a coin large number of times than we expect approximately 50% of the time head and 50% of the time tail. Table 1 shows the simulation of coin toss from number of tosses equal to 2 to number of tosses equal to 5,000 using Excel (Tools? Data Analysis? Random Number Generators).

As can be seen in table 1, as the number of trials increases, the differences between the number of actual and expected successes (heads) tends to grow, but the difference between the percentage of actual and expected successes tends to decrease. For 2 tosses, the difference between the number of actual and expected heads is 0, and for 2,000 tosses, it is 30. For 10 tosses, the difference between the percentage of actual and expected heads is 10. 00%, and for 5,000 tosses, it is equal to 0. 02%. It is not true that if we flip a coin 1,000 times we will get heads exactly 500 times.

However, it is true that if we flip a coin 1,000 times we will get heads approximately 500 times because in this case we can apply the Law of Large Numbers. It is not true that if we get tails 3 times in a row than our chances of getting heads on our next toss is greater than 50%. The reason for this is that all tosses are independent events. The previous 3-toss result will have no effect on future tosses. Therefore, if we get tails 3 times in a row than our chances of getting heads on our next toss will be again 50%.