Suppose events A and B are assumed to be independent, but when calculated, P(A and B)
eq P(A) × P(B). What does this suggest?

Question

Suppose events A and B are assumed to be independent, but when calculated, P(A and B) 
eq P(A) × P(B). What does this suggest?

BlackTom AI · Accepted Answer

The question presents a situation where A and B are assumed to be independent, yet the computed P(A and B) does not equal P(A) × P(B).

Option under consideration: The events are not independent.

First, recall the fundamental definition: for independent events A and B, P(A ∩ B) = P(A) × P(B). If this equality fails, the pair (A, B) cannot be independent under the given probability model.

Reasoning steps:
- If P(A ∩ B) ≠ P(A) × P(B), independence is violated. This is a direct consequence of the definition of independence in probability theory. Therefore, the claim that the events are not independent follows logically from the observed mismatch.
- However, the prompt states they are assumed independent. This creates a tension: either the assumption is invalid, or the calculated probabilities are incorrect due to error sources such as rounding, misinterpretation of events, or incorrect probability estimates.
- Possible sources of discrepancy include:
  - Rounding or finite-precision arithmetic leading to small numerical differences that look like inequality but are practically negligible.
  - A misidentification of the events A and B, or a miscalculation of P(A), P(B), or P(A ∩ B).
  - Dependence that was overlooked: if A and B are not truly independent, any model assuming independence would produce inconsistent results.
  - Nonstandard definitions or conditioning on another event could alter the relationship from unconditional P(A ∩ B) to a conditional quantity.

In summary, observing P(A ∩ B) ≠ P(A) × P(B) directly contradicts independence, so the correct interpretation is that the events are not independent under the given probabilities. The presence of the initial assumption of independence signals that there is likely an error elsewhere (computation, data, or modeling) that needs to be resolved before drawing firm conclusions about the relationship between A and B.

Suppose events A and B are assumed to be independent, but when calculated, P(A and B) \neq P(A) × P(B). What does this suggest?

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