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Probability theory is a branch of mathematics concerned with analyzing the likelihood of events occurring. It's rooted in the idea that we can quantify uncertainty and make informed decisions even when outcomes are partly based on chance. In essence, probability theory allows us to calculate the likelihood of an event happening using various techniques and principles.

For instance, in the provided exercise, we determine the likelihood of a taxpayer denying taking extra deductions knowing that they indeed did it intentionally. This scenario is governed by rules like the multiplication rule for independent events and the addition rule for mutually exclusive events. These rules help in establishing relationships between different probabilities, like the probability of a single event or the joint probability of two events occurring together, denoted as \(P(A)\) and \(P(B)\), respectively.

Statistical deduction, or statistical inference, is the process of using data analysis to deduce properties of an underlying probability distribution. By drawing conclusions from data subject to random variation, we employ statistical deduction to make predictions or informed guesses about a larger population based on a sample.

In our tax deduction example, statistical deduction might involve analyzing a sample of taxpayer behavior to make broader claims about all taxpayers. Here, we're using the given probabilities from a presumably representative sample to infer the likelihood of future related events. This approach lies at the heart of statistics - using a part to understand the whole, acknowledging that there is always some degree of uncertainty involved.

Bayes' theorem is a powerful formula used in probability theory and statistics to determine conditional probabilities, which are probabilities of an event occurring given that another event has occurred. The theorem uses prior knowledge or evidence to update the probability of an event.

In mathematical terms, Bayes' theorem is expressed as \(P(A|B) = \frac{P(B|A)P(A)}{P(B)}\), where \(P(A|B)\) is the probability of A given B, \(P(B|A)\) is the probability of B given A, and \(P(A)\) and \(P(B)\) are the probabilities of A and B independently. While Bayes' theorem is not used explicitly in the initial problem given, understanding it can aid in grasping the concept of conditional probability, which is fundamental when calculating something like the likelihood of a taxpayer denying cheating given they did cheat. Think of Bayes' theorem as a tool to refine our beliefs in the light of new evidence - a vital skill in both statistics and real-world decision-making.