Let's assume that we have two Bernoulli random variables: $A$ (can be true or false) and $B$ (can be true or false), and further assume we have been given $P(A=\text{true}\mid B=\text{true})$ and $P(A=\text{true}\mid B=\text{false})$.

Is it possible to calculate $P(A=\text{false}\mid B=\text{true})$ and $P(A=\text{false}\mid B=\text{false})$ from this? I think what it must hold is that these four terms must sum to one, i.e. $$P(A=\text{true}\mid B=\text{true}) + P(A=\text{true}\mid B=\text{false}) \\+ P(A=\text{false}\mid B=\text{true}) + P(A=\text{false}\mid B=\text{false}) = 1.$$