How does the symmetry of 'A and B' lead to Bayes' theorem?
Conditions
- The argument uses commutativity of logical conjunction for events.
- Both A and B have positive probability for the ordinary conditionals used here.
Reasoning, step by step
- Observe that the event 'A and B' is the same no matter which event is mentioned first.
- Note that the probability of this joint event can be computed as (B|A) or as (A|B).
- Because both expressions represent the same joint probability, they must be equal: (B|A) = (A|B).
- Divide both sides by to isolate P(A|B), giving P(A|B) = (B|A)/P(B).
- Divide both sides by to isolate P(B|A), giving P(B|A) = (A|B)/P(A).
Example
The video shows the equality chain (A|B) = P(A and B) = (B|A), emphasizing that the middle term is the same joint probability computed in two different orders.
Common misconceptions
- Thinking that (B|A) and (A|B) are different quantities because they are written in opposite orders; they are equal because both compute the same event probability P(A and B).
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Related questions
Starting from the equality (B|A) = (A|B), you can solve for either conditional probability by dividing by the corresponding marginal probability. Dividing both sides by isolates P(A|B), giving P(A|B) = (B|A)/P(B).
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The conjunction fallacy manifests when individuals judge the probability of a combined event (being a bank teller AND active in the feminist movement) as higher than the probability of one of its constituent parts (being a bank teller). Mathematically, the second event is a subset of the first, so .
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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 ).
The calculation assumes a population of 210 people: 10 librarians and 200 farmers. With 40% of librarians fitting the description (yielding 4 matching librarians) and 10% of farmers fitting it (yielding 20 matching farmers), there are 24 total matches.
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In this two-category example, the prior is the probability of the hypothesis 'Steve is a librarian' before seeing evidence, calculated as based on the stipulated population. The likelihood is the probability of the evidence 'fits the description' given the hypothesis is true, stipulated as 0.4.
Conditions: Hypothesis H: 'Steve is a librarian'; Evidence E: 'Fits the description'; Population assumption: 10 librarians, 200 farmers; Likelihood assumption: 40% for librarians, 10% for farmers
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