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In the context of rolling a die, a probability distribution is a function that describes the likelihood of each outcome within the sample space. For our specific exercise, we are rolling a die until a six appears for the first time. This means the probability distribution focuses on which roll a six appears. The formula, \( \left( \frac{5}{6} \right)^{n-1} \cdot \frac{1}{6} \), tells us about the likelihood of getting the first six on the \(n\)-th roll. Each roll has a success chance of \(\frac{1}{6}\), while the previous rolls (\(n-1\)) were non-successes with a probability \(\frac{5}{6}\).
This setup is known as a geometric distribution, where the trials are repeated until the first success is recorded. Understanding this type of distribution is crucial for managing probabilistic events where the timing of the first success is essential.

Infinite sample space in probability refers to an experiment where theoretically, there is no limit to the number of possible outcomes. In our die-rolling scenario, we can keep rolling indefinitely until a six appears for the first time. Hence, the number of possible rolls before the appearance of a six creates an infinite sample space. Each outcome consists of a sequence of rolls ending in a six, something like (no sixes, ... , no six, six).
When dealing with infinite sample spaces, it's vital to define events properly and understand how probabilities are distributed over these endless possibilities. In our case, although each trial is finite, the potential number of trials is infinite, thus broadening the sample space significantly.

The concept of a geometric series is central to solving the problem of summing probabilities in a geometric distribution. A geometric series is a series of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. In our exercise, the series you observe is derived from the probability distribution \( \sum_{n=1}^{\infty} \left( \frac{5}{6} \right)^{n-1} \cdot \frac{1}{6} \).
Recognize that this series is geometric with a common ratio, \(\frac{5}{6}\). The sum of an infinite geometric series can be found using the formula \( \frac{1}{1-r} \), where \(r\) is the common ratio, provided the magnitude of \(r\) is less than 1. For our scenario, applying this formula verifies that the sum of probabilities equals 1, thus confirming a valid probability distribution.

Understanding independent events is crucial when interpreting the die-rolling exercise. Independent events in probability theory are events where the outcome of one trial doesn't affect the outcome of another. In rolling a die, each roll is independent, meaning the result of one roll doesn't influence the next roll.
This independence is foundational for understanding why we can apply the formula \( \left( \frac{5}{6} \right)^{n-1} \cdot \frac{1}{6} \). Each time a die is rolled, there's a fresh start, with a \(\frac{1}{6}\) chance of landing a six, and a \(\frac{5}{6}\) chance of not. The independence ensures that the sequence (e.g., roll 4 brings the first six) is valid and conforms to the probability distribution described.