Why is Bayes' theorem true for events A and B of positive probability?
Conditions
- A and B are events in the same probability space.
- Both A and B have positive probability for the ordinary conditionals used here.
Reasoning, step by step
- Identify the joint probability P(A and B).
- Express P(A and B) as (B|A) by first taking the proportion of cases where A is true, then the fraction of those where B is also true.
- Express P(A and B) as (A|B) by mirroring the previous step with the roles of A and B exchanged.
- Equate the two expressions: (B|A) = (A|B).
- Divide by to solve for P(A|B), yielding P(A|B) = (B|A)/P(B).
- Alternatively, divide by to solve for P(B|A), yielding P(B|A) = (A|B)/P(A).
Example
The video displays the chain (A|B) = P(A and B) = (B|A), then rearranges it to P(A|B) = (B|A)/P(B) and P(B|A) = (A|B)/P(A).
Common misconceptions
- Mistaking the asymmetric-looking product formula for a genuinely asymmetric joint probability; they are equal because both compute the same event probability P(A and B).
- Overlooking the need for nonzero denominators in the rearranged formulas; to divide by or , one needs or respectively.
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