Why is the formula P(A and B) = crossed out at the beginning of the video?
Conditions
- The events A and B may be dependent.
- The general multiplication rule P(A and B) = (B|A) applies regardless of independence (assuming ).
Reasoning, step by step
- Identify the crossed-out formula: P(A and B) = .
- Recognize that this formula assumes independence between A and B.
- Note the corrected formula shown: P(A and B) = (B|A).
- Understand that P(B|A) accounts for how the occurrence of A changes the likelihood of B.
- Conclude that the general rule is necessary for dependent events, making the simple product rule insufficient in general.
Example
The video displays a complex 4x4 human figure grid where the simple product rule fails, contrasting it with simple 2x2 coin and 6x6 dice grids where independence holds. The red cross over P(A and B) = and the green check next to P(A and B) = (B|A) visually reinforce this correction.
Common misconceptions
- Believing that the probability of two events happening together can always be found by simply multiplying their individual probabilities, i.e., P(A and B) = .
- Assuming that introductory examples like coin flips and dice rolls represent real-world complexity, where dependence is common.
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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).
Conditions: Both A and B have positive probability for the ordinary conditionals used here.; The algebraic rearrangement requires the denominators and to be nonzero.
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 .
Conditions: Comparing the probability of a subset event against its superset event; Events are defined such that one is contained within the other
The event 'A and B' is logically identical to 'B and A', so any valid decomposition of its probability must agree. This symmetry forces the equality of the two product formulas: (B|A) = (A|B).
Conditions: The argument uses commutativity of logical conjunction for events.; Both A and B have positive probability for the ordinary conditionals used here.
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.
Conditions: Population consists only of librarians and farmers with a 1:20 ratio; Likelihoods are stipulated as 40% for librarians and 10% for farmers; Conditioning requires
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
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.