How is the law of large numbers different from the gambler's fallacy?

The sample mean $\overline{X}_n$ is defined as the arithmetic average of $n$ observed values produced by repeating the experiment associated with a random variable. It is calculated by summing the individual observations and dividing by the total number of observations.

Conditions: There must be $n$ observations.; The observations are summed and divided by $n$.

The video writes both $E(x)$ and $\mu$ because they represent the same quantity: the expected value of the random variable, which is also called the population mean. The speaker uses these notations interchangeably to emphasize that the sample mean converges to this fixed theoretical value.

Conditions: The random variable has an expected value.; $E(x)$ and $\mu$ are used as alternative notations for the population mean.

The speaker explicitly states that he is being informal about what "approach" or "convergence" means. He warns that the clip presents the theorem intuitively rather than through a rigorous epsilon-style definition, implying that the precise mathematical meaning of the limit is not fully detailed in this segment.

Conditions: The discussion is introductory.; The focus is on intuition rather than rigorous proof.

In this example, $\overline{X}_n$ represents the sample mean of $n$ repeated trials of the 100-toss experiment. It is the arithmetic average of the results (number of heads) from each of the $n$ repetitions.

Conditions: Repeated trials of the same experiment.; Each trial produces a numerical observation.; $n$ is the number of trials.

The expected number of heads is 50 because the expected value for a counting experiment like this is calculated as the number of trials multiplied by the probability of success on each trial. With 100 tosses and a fair coin (probability of heads = 0.5), the calculation is $100 \cdot 0.5 = 50$.

Conditions: The coin is fair.; There are 100 tosses per trial.; $X$ counts heads.