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A tree diagram is a visual tool that helps us break down complex problems by organizing information into branches. In probability, tree diagrams display all possible outcomes of an event and their associated probabilities. Each branch represents an outcome and leads to subsequent outcomes, creating a visual map of all combinations.
To read a tree diagram:

• Start from the root and follow the branches to the leaves.
• Multiply the probabilities along the paths to find the probability of combined events.

In the car theft exercise, we begin with two main branches: stolen by professionals or amateurs. Each of these branches further divides into recovery outcomes: recovered within 48 hours, after 48 hours, or never recovered. This structure helps in analyzing probabilities related to vehicles being stolen and recovered, or not.

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It is a central concept in probability and statistics, often employed to refine predictions based on new information.
The formula for conditional probability is expressed as:\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]where:

• \(P(A \mid B)\) is the probability of event A occurring given B.
• \(P(A \cap B)\) is the probability of both A and B occurring.
• \(P(B)\) is the probability of event B.
In the example, the probability that a vehicle stolen by a professional is recovered within 48 hours is a conditional probability. We already know the vehicle is stolen by a professional, thus we use the probability associated with that specific branch of the tree diagram, which was directly given as 24% or 0.24.

Bayesian analysis is an approach in statistics where probabilities are updated as new evidence is presented. It combines prior knowledge with new data, allowing for a more refined hypothesis as new information becomes available.
The essential component in Bayesian statistics is Bayes' Theorem, which can be formulated as:\[P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}\]where:

• \(P(A \mid B)\) is the probability of hypothesis A given data B.
• \(P(B \mid A)\) is the probability of data B given hypothesis A.
• \(P(A)\) is the prior probability of hypothesis A.
• \(P(B)\) is the probability of data B.

In our exercise, Bayesian thinking occurs when calculating the probability that a vehicle will never be recovered, regardless of the thief being professional or amateur. By starting with the initial assumptions (likelihood that a thief is professional or amateur) and updating these with the new evidence (probability of not recovering the vehicle given the type of thief), we arrive at a more comprehensive understanding with the accumulated probability of 39.84% that a stolen vehicle is never recovered in this city.