<Ul> <Li> </Li> <Li> </Li> <Li> </Li> </Ul> <P> In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times . According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed . </P> <P> The LLN is important because it guarantees stable long - term results for the averages of some random events . For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins . Any winning streak by a player will eventually be overcome by the parameters of the game . It is important to remember that the law only applies (as the name indicates) when a large number of observations is considered . There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy). </P> <P> For example, a single roll of a fair, six - sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability . Therefore, the expected value of a single die roll is </P>

When does the law of large numbers apply