The Law
of Large Numbers says that in repeated, independent trials with
the same probability *p* of success in each trial, the chance that
the percentage of successes differs from the probability *p* by
more than a fixed positive amount, *e* > 0, converges to zero
as the number
of trials *n* goes to infinity, for every positive *e*.
Note two things:

- The difference between the number of successes and the number of
trials times the chance of success in each trial (the expected number
of successes) tends to
**grow**as the number of trials increases. (In fact, this difference tends to grow like the square-root of the number of trials.) -
Although the chance of a large difference between the percentage of successes
and the chance of success gets smaller and smaller as
*n*grows, nothing prevents the difference from being large in some sequences of trials. The assumption that this difference always tends to zero, as opposed to this difference having a large probability of being arbitrarily close to zero, is the difference between the Law of Large Numbers, which is a mathematical theorem, and the Empirical Law of Averages, which is an assumption about how the world works that lies at the base of the Frequency Theory of probability.

The distribution of the number of successes in *n* independent
trials with probability *p* of success in each trial is
Binomial,
with parameters *n* and *p*.

The controls on this applet let you change the number of trials, the probability of success in each trial, and toggle between viewing either the difference between the number of successes and the expected number of successes, or the difference between the percentage of successes and the probability of success in each trial.