When is the joint probability equal to the product of individual probabilities?
Conditions
- Events A and B are independent.
- For the conditional equality P(B|A) = , event A must have positive probability.
Reasoning, step by step
- Recall the general multiplication rule: P(A and B) = (B|A).
- Apply the definition of independence: if A and B are independent, then P(B|A) = .
- Substitute for P(B|A) in the general rule to get P(A and B) = .
- Note that this equality is the defining condition for independence (when ).
Example
The video shows examples like coin flips and dice rolls where P(TT) = and P(One One) = , assuming independent trials. It contrasts this with the sibling heart disease example, where correlation might violate this simple product rule.
Common misconceptions
- Assuming P(A and B) = for all events; this formula only holds if events A and B are independent. Correlated events require different calculations.
- Confusing fairness with independence; fairness specifies marginals, while independence licenses multiplication.
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