Does the observed success proportion get closer to the theoretical probability $p$ at every single step of independent repeated trials?

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 law of large numbers states that the average of many trials converges to the expected value, but it does not imply that future trials will compensate for past deviations. The gambler's fallacy is the mistaken belief that if early trials deviate from the mean, later trials must produce opposite outcomes to 'balance out' the average immediately.

Conditions: The speaker is contrasting a common intuition with the actual meaning of the theorem.; Trials are independent.

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.

Comparing these different sample sizes illustrates that larger samples tend to produce observed proportions that are more stable and closer to the theoretical probability. With fewer trials (like 10), random noise can cause significant deviations from the expected proportion.

Conditions: The comparison involves sample sizes of 10, 100, and 1,000 trials.; The goal is to understand the stabilization of observed proportions.