Editing Law of large numbers (section)

==Examples==
For example, a single roll of a six-sided [[dice]] produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal [[probability]].  Therefore, the [[expected value]] of the roll is:

<math display="block"> \frac{1+2+3+4+5+6}{6} = 3.5</math>

According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the [[sample mean]]) will approach 3.5, with the precision increasing as more dice are rolled.

It follows from the law of large numbers that the [[empirical probability]] of success in a series of [[Bernoulli trial]]s will converge to the theoretical probability. For a [[Bernoulli random variable]], the expected value is the theoretical probability of success, and the average of ''n'' such variables (assuming they are [[Independent and identically distributed random variables|independent and identically distributed (i.i.d.)]]) is precisely the relative frequency.
[[File:Law_of_large_numbers_(black_%26_red_balls).png|thumb|295x295px| This image illustrates the convergence of relative frequencies to their theoretical probabilities. The probability of picking a red ball from a sack is 0.4 and black ball is 0.6. The left plot shows the relative frequency of picking a black ball, and the right plot shows the relative frequency of picking a red ball, both over 10,000 trials. As the number of trials increases, the relative frequencies approach their respective theoretical probabilities, demonstrating the law of large numbers.]]
For example, a [[fair coin]] toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to {{frac|1|2}}. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly {{frac|1|2}}. In particular, the proportion of heads after ''n'' flips will [[almost surely]] [[limit of a sequence|converge]] to {{frac|1|2}} as ''n'' approaches infinity.

Although the proportion of heads (and tails) approaches {{frac|1|2}}, almost surely the [[absolute difference]] in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips.

Another good example of the law of large numbers is the [[Monte Carlo method]]. These methods are a broad class of [[computation]]al [[algorithm]]s that rely on repeated [[random sampling]] to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.<ref>{{Cite journal|last1=Kroese|first1=Dirk P.| last2=Brereton|first2=Tim| last3=Taimre|first3=Thomas|last4=Botev|first4=Zdravko I.|date=2014|title=Why the Monte Carlo method is so important today|journal=Wiley Interdisciplinary Reviews: Computational Statistics| language=en| volume=6| issue=6|pages=386–392|doi=10.1002/wics.1314|s2cid=18521840}}</ref>