91Ó°ÊÓ

The concept of the complement rule is fundamental in understanding probabilities. Essentially, the complement rule dictates that the probability of an event not occurring is equal to one minus the probability of the event occurring. To put it simply, the complement gives us everything that our original event does not.
For example, if we want to determine the probability of not getting a digital book, we look at the event where a digital book is indeed purchased, which has a given probability, and then subtract that from 1 (which represents the certainty of any outcome happening). In mathematical terms, if we denote the probability of purchasing a digital book as \(P(Digital)\), then the probability of not purchasing a digital book, or its complement \(P(Not\ Digital)\), is calculated as:
\[P(Not\ Digital) = 1 - P(Digital)\].
This rule is immensely useful as it allows us to quickly find the probability of the opposite of any given event, as long as we know the probability of the event itself.

Disjoint events, also known as mutually exclusive events, refer to events that cannot occur at the same time. Understanding these types of events is crucial when calculating probabilities because it helps us determine when we can simply add probabilities together for an either/or scenario.
In the given example, purchasing a digital book and purchasing an audio book are disjoint events. A book cannot be both digital and audio at the time of purchase; it must be one or the other. Therefore, when asked about the probability of purchasing either a digital or audio book, we can add the individual probabilities of these disjoint events to get the total probability.
Here’s how it's applied in our exercise:
\[P(Digital\, or\, Audio) = P(Digital) + P(Audio)\].
Remember, this addition rule only works for disjoint events since their occurrences do not overlap. In cases where events can occur simultaneously, other rules of probability must be applied.

The probability of combined events becomes relevant when we have more than one outcome that we consider a 'success' for an event. In our bookstore scenario, we're looking at combined events when we calculate the probability of purchasing a printed book, which can either be in hardcover or paperback format. These are also examples of disjoint events—in this situation, a book cannot be both hardcover and paperback, so we calculate the probability of either event by adding the probabilities of hardcover and paperback together.
Mathematically, we express this as:
\[P(Printed\ Book) = P(Hardcover) + P(Paperback)\].
Combined events probabilities help us understand the likelihood of one event or another occurring when we have multiple outcomes that we're interested in. This extends beyond disjoint events to include events that can happen at the same time, though the calculation becomes more complex as we must avoid counting overlapping probabilities twice. For example, if someone was interested in buying a book that could be either paperback or digital, and these were not mutually exclusive (such as a bundle offer), we would have to adjust our calculation to accommodate the overlap.