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Understanding the sample space is foundational in the study of probability. It encapsulates all the potential outcomes of a probability experiment. In the context of our delivery options scenario, where a department store offers four distinct methods for receiving online orders, the sample space consists of these four options: expedited overnight delivery, expedited second-business-day delivery, standard delivery, and delivery to the nearest store for customer pick-up.

Each of these options is known as a 'simple event'—an outcome that cannot be broken down any further. The sample space is important because it sets the stage for determining the likelihood of any combination of events, such as the chance of a customer selecting any specific delivery option. Recognizing the full range of possibilities is crucial when calculating probabilities, which we will explore further in the subsequent sections.

Calculating probabilities involves determining the chances of various events occurring within our defined sample space. The probabilities of all simple events in a sample space should sum up to 1, representing the certainty that one of the events will occur. Let's take a closer look at the delivery example. Here, we calculated the probability of the pick-up delivery option by subtracting the sum of the known probabilities from 1:

• \(P(\text{Pick-up}) = 1 - P(\text{Overnight}) - P(\text{Second-day}) - P(\text{Standard}) = 1 - .1 - .3 - .4 = .2\)

This calculation highlights the role of the complementary events in probability. To find the odds of expedited delivery, a composite event, we simply summed the probabilities of the overnight and second-business-day delivery options:

• \(P(\text{Expedited}) = P(\text{Overnight}) + P(\text{Second-day}) = .1 + .3 = .4\)

Similarly, for a compound event comprising the standard and pick-up delivery options, their individual probabilities were added together:

• \(P(\text{Standard or Pick-up}) = P(\text{Standard}) + P(\text{Pick-up}) = .4 + .2 = .6\)

Such calculations illustrate the additive nature of probabilities when considering mutually exclusive events, and they underscore the importance of accurately identifying all aspects of the sample space.

Probability theory is the mathematical framework that deals with calculating the likelihood of different events. It is based on a set of axioms, which together with an understanding of sample spaces and event probabilities, provide the tools needed to evaluate random phenomena across disciplines.

Basic principles include concepts like mutually exclusive events—events that cannot occur simultaneously. When events are mutually exclusive, like the delivery options in our example, the probability of any one of them occurring is the sum of their individual probabilities if there's no overlap.

The opposite would be non-exclusive events, where outcomes can occur in tandem. In such cases, probability calculations must adjust for the overlap. Probability theory also deals with independent events (one event’s occurrence doesn't affect another's) and dependent events (one event’s outcome influences another's possibility).

Using probability theory, we can elucidate the uncertainty that comes with various real-world scenarios, whether it's in choosing delivery options or predicting weather patterns, helping us make more informed decisions amidst uncertainty.