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Randomness refers to the occurrence of outcomes in a manner that is unpredictable and shows no recognizable pattern. In the context of a casino or any game which claims to function randomly, this implies that each event or decision is spontaneous and free from pre-established sequences.
For example, when we say a machine is truly random, it means every action taken by the machine results independently without being influenced by the previous actions. For machines like video roulette, randomness means that each spin is an isolated event. The outcome of a spin doesn't carve the pathway for the next, making predictions based solely on past events impossible.
This independence of events is crucial to ensure fairness, as it prevents any biases from skewing the result towards a particular pattern or outcome over time, maintaining an element of surprise and chance for each play.

Independent events are a critical aspect of understanding randomness. In probability, two or more events are considered independent if the occurrence of one does not affect the occurrence of the other.
Using the context of the video roulette machine, if each spin on the machine is independent, the result of the previous spin does not influence or alter the probability of the next spin. Whether the previous spin landed on red or black, it doesn't make any particular outcome more likely in the subsequent spins.
Maintaining independence between spins is crucial in upholding the claim of true randomness as posited by the casino. Players rely on the game's independence to ensure every spin offers a fair chance, thus upkeeping the integrity and trust in the game's processes.

Roulette probability is the fundamental principle which dictates the odds of landing on any particular number or color on the roulette wheel. For standard American roulette, with numbers from 1 to 36, as well as 0 and 00, the total number of possible outcomes is 38.
Each number has an equal chance of being the result, giving it a probability of \( \frac{1}{38} \) per spin. This probability holds if the roulette is truly random and independent for every spin played by a participant.
At its core, this probability ensures the game remains balanced. Over a large number of spins, if one were to observe the outcomes, the distribution of every number chosen should be approximately equal. Any deviation could suggest a breach in the randomness and independence of the system, questioning the fairness of the game's structure and operation.