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Understanding the probability of an event is crucial for grasping the basics of probability theory. Probability quantifies the likelihood that a specific event will occur within a set of conditions. It’s calculated by considering the ratio of the favorable outcomes to the total number of possible outcomes.

Let's consider an event, say 'E', within a larger set of possible outcomes — the sample space 'S'. If we know that event 'E' can happen in 'a' different ways, and there are 'b' total possible outcomes in the sample space, we express the probability of the event 'E' as:
\[ P(E) = \frac{a}{b} \]
If the probability of 'E' is close to 1, it means the event is very likely to occur, while a probability close to 0 indicates it is very unlikely to happen. In any case, probabilities will always fall between 0 and 1, inclusive.

The complement of an event offers another perspective to understanding the likelihood of outcomes. The complement of an event 'E' is denoted as \(E^c\) and includes all possible outcomes in the sample space that are not part of 'E'.

The probability of the complement of 'E', denoted \(P(E^c)\), tells us the likelihood of everything that doesn't satisfy the conditions of 'E' occurring. It's essentially like flipping the scenario: instead of looking for how likely 'E' is to occur, we're exploring the odds of it not occurring at all.
Given the probability of event 'E', we calculate the complement using the formula:\[ P(E^c) = 1 - P(E) \]
This formula makes intuitive sense because if we add the probabilities of an event and its complement, we should account for all possible outcomes, which sums to 1.

In probability, when we're dealing with two events, we often want to know how likely they are to occur together. This is where the probability of intersection comes into play. It’s denoted as \(P(E \cap F)\) and describes the probability of events 'E' and 'F' both happening.

To find this probability, one must consider only the outcomes that are common to both events. It's an important concept because it allows us to evaluate the connections and interactions between events. For example, if 'E' represents 'having a sandwich' and 'F' means 'having soup', then \(P(E \cap F)\) would correspond to the likelihood of having both a sandwich and soup together.
When events are independent, their intersection probability is the product of their separate probabilities. However, events often affect each other, meaning we need to account for the overlap in their occurrence, as seen in the formula provided earlier:\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
This formula for the probability of the union of two events includes subtracting the intersection to avoid double-counting the cases where both events happen together.