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The formula P(A and B) = $P(A)P(B)$ is crossed out because it is only valid for independent events. The video emphasizes the general multiplication rule, P(A and B) = $P(A)P$(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) = $P(A)P$(B|A) applies regardless of independence (assuming $P(A) > 0$).

The formula P(A and B) = $P(A)P(B)$ is crossed out because it is only valid for independent events. The video emphasizes the general multiplication rule, P(A and B) = $P(A)P$(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) = $P(A)P$(B|A) applies regardless of independence (assuming $P(A) > 0$).

The values $P(A\cap B)=0.0225$ and $P(B)=0.5$ are described as illustrative inputs because they are hypothetical numbers chosen to demonstrate the calculation method, rather than verified population statistics. This distinction prevents learners from misinterpreting the example as factual demographic data.

Conditions: The context is a pedagogical example in a video.; The events involve alcohol dependence and gender.

The values $P(A\cap B)=0.0225$ and $P(B)=0.5$ are described as illustrative inputs because they are hypothetical numbers chosen to demonstrate the calculation method, rather than verified population statistics. This distinction prevents learners from misinterpreting the example as factual demographic data.

Conditions: The context is a pedagogical example in a video.; The events involve alcohol dependence and gender.

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.

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.

Swapping the events changes both the numerator and the denominator in the conditional probability formula. $P(A\mid B)$ uses $P(A \cap B)$ as the numerator and $P(B)$ as the denominator, whereas $P(B\mid A)$ uses $P(A \cap B)$ as the numerator but $P(A)$ as the denominator.

Conditions: Events A and B have different marginal probabilities ($P(A) \neq P(B)$).; Both events have positive probability.

Swapping the events changes both the numerator and the denominator in the conditional probability formula. $P(A\mid B)$ uses $P(A \cap B)$ as the numerator and $P(B)$ as the denominator, whereas $P(B\mid A)$ uses $P(A \cap B)$ as the numerator but $P(A)$ as the denominator.

Conditions: Events A and B have different marginal probabilities ($P(A) \neq P(B)$).; Both events have positive probability.