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When discussing randomness in statistics, the concept of probability is foundational. Probability is a measure that quantifies the likelihood that a specific event will occur. Say you're flipping a fair coin; the probability of getting heads is 50%, because there are two equally likely outcomes: heads or tails.

In the context of a roulette wheel, each slot has a probability associated with the ball landing in it. The more slots there are, the lower the chance for the ball to land in any specific one. For a standard European roulette wheel, with 37 slots (1 to 36 and single '0'), the probability of the ball landing in a specific slot is roughly 2.7%. This assumes that the wheel is fair, meaning each slot has an equal chance of being landed on. The act of spinning the wheel and releasing the ball should, ideally, be under conditions that do not bias these probabilities.

Random distribution refers to the way outcomes are spread out over a range of possibilities. In a perfectly random distribution, every outcome has an equal chance of occurring, and the outcomes do not affect each other. This can be also described as having 'independent and identically distributed' outcomes.

For our roulette example, envision each spin of the wheel as independent from the last; the ball's landing on a particular number once does not make it more or less likely to land on that same number again. This independence is what fuels the concept of randomness in games of chance. A large number of spins would show the ball landing on each number roughly an equal number of times, exhibiting what statisticians call a 'uniform distribution' over the long term.

The notion of roulette wheel randomness centers around the unpredictability of the spin's outcome. When a casino claims that its roulette wheel is 'truly random,' it suggests that every spin's result is based solely on chance and that each number has an equal probability of being the outcome, with no predictable pattern or influence affecting it.

In practice, there are many factors that can compromise the randomness of a roulette wheel, such as a wheel bias or a dealer's specific spinning technique. However, sophisticated casinos closely monitor and regularly calibrate their roulette wheels to ensure that the randomness is maintained. This strict adherence to random distribution is essential to guarantee the fairness and integrity of the game, connoting that neither the house nor the players can predict or influence the results of a spin.