Average convergence, not repayment
For repeated independent fair coin flips, the fraction of heads tends towards one-half as the sample grows. After an early streak of heads, later flips need not favour tails. Ordinary balanced additions can make the early imbalance a smaller fraction of the total. The next flip's probability remains unchanged.
Similarly, a theoretical game return can describe a long-run average under its model. A player is not guaranteed to receive that percentage over a chosen number of rounds. Large rare prizes and variable outcomes can produce substantial differences in shorter records.
Percentage precision and money exposure
A proportion can become closer to its model while the absolute monetary result remains a loss. At a negative expected return, increasing turnover increases modelled expected loss. Pursuing convergence by continuing to stake money does not transform the expectation into profit or create a reimbursement rule.
Assumptions and changing configurations
The observations need the relevant conditions, such as appropriate independence and a stable distribution for a simple version of the result. Mixing game versions, stakes or promotional rules can change the quantity being averaged. A growing dataset is not automatically comparable when its underlying process changes halfway through.
How to use the concept in research
State the model, observation unit, sample size and distribution before drawing conclusions. Report observed return as observed, and theoretical return as theoretical. The phrase law of averages is often used loosely to imply that luck must balance immediately; that is not the mathematical result. The theorem explains average behaviour under assumptions, not what a particular person's next round or bank balance will be.
Does a larger sample always get closer at every step?
No. Convergence is not a requirement that the average improves monotonically after each observation. A new result can move it farther from the expected value even as the long-run mathematical statement remains valid under its assumptions. Likewise, a session can temporarily move towards an advertised return and later move away. Do not treat one favourable movement as evidence that recovery is underway. Examine the model and sample rather than turning a convergence theorem into a prediction about the next account balance.