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At the heart of understanding random events and making predictions about them lies probability theory. It's a branch of mathematics that measures the likelihood of occurrence of different events. For instance, when tossing a fair coin, probability theory helps us determine the chances of landing on heads or tails.

Applying probability theory involves defining the sample space of an experiment, which is the set of all possible outcomes. In the case of our coin toss experiment, the sample space includes all the sequences we could get by tossing a coin three times. Knowing the sample space is essential because the probability of any event is calculated as the number of favorable outcomes divided by the total number of outcomes in the sample space.

Understanding the outcomes of a coin toss is a fundamental example of studying probability. When a coin is tossed, we often assume it's fair, meaning it has an equal chance of landing on heads or tails. For a single toss, the sample space is simple: {H, T}.

However, when we repeat this toss multiple times, as in our exercise of tossing it three times, things get more interesting. We apply the rules of combinatorics as well as incorporate an understanding of independent events—the result of one toss doesn't affect the next. The sample space exponentially grows with each additional toss. Instead of 2 outcomes, after three tosses, we've expanded to 8 potential outcomes. These outcomes provide a visual representation of all possible events, which in turn can be used to calculate probabilities for different combinations of heads and tails.

Combinatorics is a field of mathematics primarily concerned with counting, arrangement, and combination of items. In probability, combinatorics plays a key role in determining the size of the sample space and calculating the probabilities of various events.

For example, the fundamental counting principle, illustrated in our coin toss experiment, allows us to determine the total number of outcomes by multiplying the number of choices at each stage. Thus, with two outcomes possible for each of three coin tosses, we have a total of 2 x 2 x 2, which is 8 possible outcomes. As we delve deeper into combinatorics, we encounter other concepts like permutations and combinations, which aid in solving more complex probability problems where we have to account for order or where specific items are selected from larger sets.