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Understanding the concept of complementary events is fundamental in probability theory. Complementary events are two outcomes of an experiment that are mutually exclusive and exhaustive. This means that they cannot both happen at the same time, and one of the events must occur whenever the experiment is performed. A common example of complementary events is flipping a coin: the coin will either land on heads or tails, not both, and there is no third possible outcome.

For any event A, the complementary event is often denoted as A'. If the probability of A occurring is P(A), then the probability of A not occurring - that is, A' happening - will be 1 minus P(A), or P(A') = 1 - P(A). This rule must always be true, as the sum of the probabilities of complementary events always equals one. Understanding this relationship is crucial when a question asks for the probability of an event not happening, as seen in the exercise above where calculating that event A does not occur is as simple as subtracting P(A) from 1.

The sample space is another key concept in probability. It represents all possible outcomes of a random experiment, and is often denoted by the symbol S. Determining the sample space is an essential first step in any probability calculation because it sets the foundation for determining the likelihood of various events. For instance, when rolling a six-sided die, the sample space includes the set {1, 2, 3, 4, 5, 6}, as these are all the potential outcomes.

In the context of the exercise, the sample space is defined by the inclusive range of integers from 16 to 28. Counting inclusively means that both 16 and 28 are part of the sample space, hence there are 28 - 16 + 1 = 13 possible outcomes. In probability calculations, a well-defined sample space ensures that all possible outcomes are accounted for when determining probabilities.

Finally, we have probability calculation, which involves using the principles of probability to determine the likelihood of different events. Probabilities are always expressed as numbers between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. To calculate the probability of an event, one typically divides the number of ways an event can occur by the total number of possible outcomes in the sample space.

For example, let's consider the probability of rolling a 3 on a standard die. There is one way to get a 3 among the six possible outcomes, so the probability is 1/6. In the second part of the exercise, we use the rule of complementary events for probability calculation. Given that the probability of selecting an integer less than 20 is 0.78, we subtract this from 1 to find the probability of selecting 20 or more. This is a critical skill in probability - using known probabilities to calculate unknown ones, particularly with complementary events. This type of reasoning helps solve a variety of real-world and mathematical problems.