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Probability theory is the branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between 0 and 1. A probability of 0 indicates that the event cannot occur, while a probability of 1 means the event is certain to happen.

When rolling two fair dice, we regard each roll as an independent event because the outcome of one roll doesn't affect the outcome of the other. Probability theory helps us determine how likely it is to roll a die and see a number that is exactly twice the number on the other die. By using a probability formula, which is the ratio of favorable outcomes to the total possible outcomes, we can predict the chances of such an event occurring.

Random experiments are processes by which we observe outcomes that arise by chance. In our context, rolling dice is a perfect example of a random experiment, since we cannot predict with certainty what number will appear on each die. There is an element of unpredictability, but the randomness is uniform as each outcome has an equal chance of occurring—assuming the dice are fair.

Each roll of a die can result in one of six outcomes. When dealing with two dice, we consider the combination of both their outcomes. The concept of a random experiment underpins probability theory and provides a framework for understanding and calculating the probability of complex events.

Combinatorics in probability is the study of counting, ordering, and arrangement that helps in analyzing complex problems where there are many possible combinations. When rolling two dice, combinatorics is used to systematically enumerate all possible outcomes.

For instance, we identified the specific combinations where one die shows twice the value of the other. These combinations are part of the sample space that we consider 'successful' in the context of our specific event. Combinatorics not only allows us to count these successful combinations but also the total combinations that two dice can produce. By applying combinatorial techniques, we can assure that no possible outcome is left uncounted.

Sample space analysis involves listing and evaluating all the possible outcomes of an experiment. The sample space for rolling a single die is straightforward—six possible outcomes: 1, 2, 3, 4, 5, or 6. However, with two dice, the sample space becomes larger because we have to consider the result on both dice, leading to 6 x 6, which is 36 possible outcomes.

With our exercise, we need to extract from the total sample space the relevant outcomes where one die's number is twice the other. After identifying these 6 specific cases, we divide them by the total number of outcomes, giving us the probability. It's essential that the sample space is well-defined and complete for accurate probability calculations. By thorough sample space analysis, we assure a clear understanding of all the scenarios that can possibly arise from our experiment.