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At the core of understanding various outcomes of events is probability theory. This branch of mathematics deals with the quantification of the likelihood that an event will occur. It is fundamental to various fields such as statistics, finance, science, and many forms of decision-making.

When we express probability, it typically ranges between 0 and 1, where 0 indicates an impossible event, and 1 signifies a certain event. In the context of the exercise, for example, the probability of a family owning a pet (whether a dog, a cat, or both) is a value between 0 and 1 based on the percentage of families that own pets within the community.

Understanding how to convert percentages and proportions into probabilities is essential. To transition from a percentage to a standard probability, we divide by 100. For instance, if 36 percent of families own a dog, this is expressed in probability as \(0.36\) or \(\frac{36}{100}\).

Joint probability is the probability that two events will occur simultaneously. In probability notation, it's denoted by \(P(A \cap B)\), where \(A\) and \(B\) are two events. This probability informs us about the occurrence of event \(A\) and event \(B\) at the same time.

In the given exercise, the joint probability to find is the likelihood that a family owns both a dog and a cat. This is represented as \(P(Dog \cap Cat)\). Mathematically, if we know the individual probabilities of each event and the conditional probability that a family owning a dog also owns a cat, we can determine the joint probability. It's calculated by multiplying the probability of one event by the conditional probability of the other event occurring given that the first event has occurred, formulated as \(P(Dog) \times P(Cat | Dog)\).

This joint probability concept is vital in various real-world applications, such as understanding the likelihood of concurrent weather events or the risk of simultaneous system failures in engineering.

Bayes' theorem is a powerful result in probability theory that allows us to update our probability estimates based on new information. It is expressed as \(P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}\), where \(P(A | B)\) is the probability of event \(A\) occurring given that event \(B\) has occurred.

In the exercise's context of conditional probability, Bayes' theorem applies to part (b), which queries about the probability of a family owning a dog given that it already owns a cat \(P(Dog | Cat)\). With the established joint probability \(P(Dog \cap Cat)\) and the probability of owning a cat \(P(Cat)\), Bayes’ theorem helps to unravel this conditional probability.

Breaking it down with the numbers from the exercise, we calculate this conditional probability by taking the joint probability that a family owns both pets and dividing it by the probability of owning a cat, resulting in \(P(Dog | Cat) = 0.264\) or 26.4%. Bayes' theorem has a vast array of uses beyond textbook exercises, including the fields of machine learning, diagnostic testing, and decision-making processes.