The law of large numbers simply explained
The law of large numbers plays an important role in stochastics and is used in many areas of everyday life (for example in insurance or in physics). But what is actually behind this graceful sounding term and how can you easily understand it?
Introduction to the law of large numbers
The law of the great Counting The easiest way to understand it is to use a particularly simple example. In a simple dice roll with a fair dice, there are six different outcomes (the numbers 1 to 6), all of which have the same probability. For example, P ("6 thrown") = 1/6. But what does this have to do with the law of large numbers?
- Suppose you run this random experiment 100 times under the same circumstances and make a tally How often the numbers 1 to 6 occurred, then you have determined the absolute frequencies in this way. If you put this in relation to the number of dice rolls, you will get the relative frequencies. If you have 100 throws e.g. B. If the six were thrown 20 times, the relative frequency of the six would be 20/100 = 1/5. The actual probability of rolling a six is not 1/5, but 1/6.
- The law of large numbers now says that the more often you do the random experiment among the same Repeating circumstances, the closer the relative frequency of the random outcome approaches the Probability at. In between, the relative frequency can of course also differ further from the probability if, for example, in the dice roll example, you hit a 6 100 times in a row in the meantime roll the dice. In the long run, however, the two sizes will converge.
- You should not interpret this principle by betting on red in roulette just because the last 10 rounds were always black. Even if the number 25 has been drawn the most frequently in the "6 out of 49" lottery so far, this does not mean that this number will be drawn less often in the future! In poker, too, you shouldn't just "all-inn" a flush draw on the flop just because you got the Flush didn't hit the last five all-ins after the flop and yes he'll come at some point got to". The random experiments are independent of one another and the different results are always equally likely. Or in short: what was in the past has no effect on the future.
- This law is in the mathematics divided into a weak law for large numbers and a strong law for large numbers.
The calculation of probability is one of those types of mathematics that one ...
Mathematical explanation of the strong and weak law
- In the weak law of large numbers, you have Yi with i∈N given as real random variables which all have the same expectation µ. Furthermore, two different random variables are uncorrelated. Now you determine the arithmetic mean of n of these random variables, so you get Yn'= (Y1+ Y2+... + Yn) / n. Now form the limit for n towards infinity, then for all ε> 0: limn-> ∞ P (| Yn'-µ | n')n∈N converges stochastically to µ with increasing sample size N.
- With the strong law of large numbers, you gave the same starting values. Now, however, P (limn->∞ Yn'=µ) = 1. The strong law of large numbers is thus formulated even more narrowly, it even implies the weak law of large numbers (if the large law is fulfilled, then the small law is also fulfilled. However, the reverse does not apply).
As you can see, the law of large numbers is a fundamental building block of the statistics and indispensable. In the physics For example, the law of large numbers plays an important role. Do you have to deal with a huge number of measurements that have to be carried out over and over again under the same circumstances and deviates If the measurement result always clearly goes up, then the probability is high that a systematic error is present.
