The only explanation I could come up with is that when using Bayes Theorem, the denominators don't match up.
But is there a more intuitive explanation that can be verbalized?
2026-04-25 21:26:57.1777152417
Why is P(A |B) + P( A |B') ≠ P(A)?
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For example take 2 coin flips, the two events are $A=$'The first coin flip is heads' and $B=$'The second coin flip is heads'.
Then $P(A|B)=P(A|B^{C})=P(A)=0.5$, but clearly $0.5+0.5\neq0.5$.
There is no reason that these two probabilities should add up to $P(A)$, conditional probability is very different from the probability of an intersection.