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The formula P(A and B) = is crossed out because it is only valid for independent events. The video emphasizes the general multiplication rule, P(A and B) = (B|A), which works for both independent and dependent events.
Conditions: The events A and B may be dependent.; The general multiplication rule P(A and B) = (B|A) applies regardless of independence (assuming ).
In the geometric model, the total sample space is represented as a large square. The prior probability determines the width of a vertical strip representing the hypothesis, and the likelihood determines the height of the shaded region within that strip where the evidence occurs.
Conditions: Total sample space is normalized to unit area; Events H and E are treated as measurable subsets
Critics point out that priors are subjective starting points influenced by background knowledge. The definition of the source population matters significantly: whether Steve is drawn randomly from national census data or is someone personally known to the observer changes the baseline widths in probability diagrams.
Conditions: Application of base rates to specific individual profiles; Ambiguity exists regarding the sampling frame (random vs. personal acquaintance)
Bayes' theorem is true because the joint probability of two events can be decomposed in two symmetric ways. The probability that both A and B occur is equal to the probability of A multiplied by the conditional probability of B given A, and also equal to the probability of B multiplied by the conditional probability of A given B.
Conditions: A and B are events in the same probability space.; Both A and B have positive probability for the ordinary conditionals used here.
Framing problems using natural frequencies (e.g., counts out of a fixed population like 210 people) aligns with findings from psychologists Kahneman and Tversky. Concrete numbers of people make complex conditional relationships tangible, reducing cognitive load.
Conditions: Problems involve conditional probabilities or Bayesian inference; Comparison is between natural frequency framing and percentage/probability framing