Why is the conditional probability of exactly two empty boxes given at least one empty box written as P(B|A) = P(AB)/P(A)?
Conditions
- The experiment involves placing 4 distinct balls into 4 distinct boxes.
- A is defined as "at least one box is empty".
- B is defined as "exactly two boxes are empty".
- The formula requires .
Reasoning, step by step
- Identify the condition event A and the target event B from the problem statement.
- Translate the textual requirement into the conditional probability notation P(B|A).
- Apply the definition of conditional probability to express P(B|A) as P(AB)/P(A).
- Calculate and P(AB) separately to find the final ratio.
Example
The instructor labels A and B below the problem statement and writes P(B|A)=P(AB)/P(A) on the screen.
Common misconceptions
- Confusing the joint probability P(AB) with the conditional probability P(B|A).
- Assuming P(B|A) = without checking if A and B are independent (they are not, as B implies A).
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