While 'random experiment' is the standard statistical term, the provided answer key identifies 'Variable'. In some contexts, a variable is used to describe the changing nature of outcomes. We preserve the answer key while noting that 'Random experiment' is the more precise technical definition.

Since the sum of probabilities for all possible outcomes must equal 1, we have P(H) + P(T) = 1. Given P(T) = 2P(H), we substitute this into the equation: P(H) + 2P(H) = 1. This simplifies to 3P(H) = 1, resulting in P(H) = 1/3. Therefore, the probability of heads is one-third.

In standard probability theory, events are typically denoted by uppercase Latin letters such as A, B, or C. These represent subsets of the sample space. While other notations exist, capital letters are the conventional standard in introductory and advanced probability textbooks.

The probability of an event is the ratio of the favorable outcomes to the total outcomes. Since 30% of the fruits are oranges, the probability of selecting an orange is 30/100, which simplifies to 0.3.

The simultaneous occurrence of two events A and B is denoted by the intersection of the two sets, written as A ∩ B. This represents the set of all outcomes that are common to both event A and event B. It signifies that both events must happen at the same time.

Complementary events are two mutually exclusive outcomes that represent all possible results in a sample space. Since the sum of their probabilities must equal one, if one event does not occur, the other must occur, making them perfect complements in binary scenarios.

The classical approach to probability, also known as the a priori approach, is based on the assumption that all outcomes in the sample space are equally likely. Under this framework, the probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. This method is typically applied to games of chance, such as rolling dice or flipping coins, where symmetry dictates outcome likelihood.

To find the probability of the union of two events, we identify outcomes where at least one die is 4: (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4) [11 outcomes]. Outcomes where the sum is 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) [6 outcomes]. The intersection (3,4) and (4,3) occurs twice. Using the inclusion-exclusion principle: 11 + 6 - 2 = 15. Thus, 15/36.

Probability theory encompasses various ways to relate events. Joint probability concerns the intersection of events, marginal probability concerns individual events, and conditional probability concerns the likelihood of an event given another. All these concepts are fundamental to understanding the behavior of independent and dependent events.

Probability is defined as the measure of the likelihood that an event will occur. It is not a 'collection of experiments'; rather, it is a numerical value assigned to the outcome of a random experiment. Options A, B, and C are standard, correct properties of probability theory.