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Probability theory is a fundamental branch of mathematics that helps us understand the likelihood of different outcomes. It's all about predicting the future with numbers. By working with probabilities, we can make informed decisions based on expected outcomes rather than mere guesses.
For any event, the probability can range from 0 to 1, where 0 means the event cannot happen, and 1 means it certainly will. The probabilities of all possible outcomes of a random experiment always add up to 1.
The essence of probability theory is to deal with uncertainty in the world. For example, what are the chances it will rain tomorrow? That's where probability steps in to help us calculate and understand these chances better.

When dealing with events in probability, it's crucial to understand unions and intersections. - **Union (\(A \cup B\))**: This represents "either A or B or both happening". To calculate the probability of a union of two events, you add their individual probabilities and subtract the probability of both events happening together. The mathematical formula is \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]. Using this formula, we can determine that the likelihood of having at least one type of credit card in the example is 0.65.
- **Intersection (\(A \cap B\))** refers to "both events happening together". In our case, it represents a student having both Visa and MasterCard, with a probability of 0.25.
These operations help us understand more complex scenarios where multiple conditions are involved.

The complement rule is a handy tool in probability that helps figure out the likelihood of an event 'not' happening. This rule states that the probability of the complement of an event is equal to 1 minus the probability of the event itself. In formula terms, it's expressed as \( P(A') = 1 - P(A) \).
In our example, after determining \( P(A \cup B) = 0.65 \), we use the complement rule to find the probability of neither event happening, represented as \((A \cup B)'\). The calculation goes \[P((A \cup B)') = 1 - 0.65 = 0.35\].
This means there's a 35% chance that a selected student has neither type of credit card. The complement rule is great because it simplifies problems where direct calculation might be more challenging.

Events are called mutually exclusive if they cannot happen at the same time. When two events are mutually exclusive, the probability of both occurring simultaneously is zero, i.e., \( P(A \cap B) = 0 \).
However, in the provided example, we see that \( P(A \cap B) = 0.25 \), which means that A and B are not mutually exclusive. Both can happen simultaneously, as they aren't independent of each other.
Understanding whether events are mutually exclusive can simplify the calculation of probabilities since, for mutually exclusive events, \(P(A \cup B) = P(A) + P(B)\). When they aren’t mutually exclusive, the intersection probability must be subtracted, as shown in the example.