# Our intuition can be mathematical

Sometime maths can be quite an intuitive thing. The Law of Large Numbers is such a case.

## Law of Large Numbers (LLN)

The LLN tells us that when the number of trials increases, the sample mean approaches the population mean. Or in more mathematical terms: or

is the sample mean and is calculated as following:

## How big does n have to be?

How big n needs to be depends on how big a and p are. Nonetheless in a lot of cases n can be smaller than one might expect. We can demonstrate this with some lines of python code:

from scipy.stats import binom import matplotlib.pyplot as plt import numpy as np #Show that when n increases the probability that abs(X-mu) < a converges 1 for a in np.arange(0.1,0.01,-0.01): list_n = [] list_p = [] p = 0.5 n = 10 while (binom.cdf(int((p+a)*n), n, p)-binom.cdf(int((p-a)*n), n, p)) < 0.99999: list_p.append(binom.cdf(int((p+a)*n), n, p)-binom.cdf(int((p-a)*n), n, p)) list_n.append(n) n += 10 plt.plot(list_n, list_p, label='a={}, p={}'.format(round(a,2),p)) plt.legend() plt.show()

The above code produces the following output:

We can see that for a probability of 0.5 and a=0.1, n just needs to be around 500.

The LLN has a lot of areas where it can be applied, for example in economics, in finance and in insurance. We will shortly come to a quite similar theory, the central limit theorem.

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