Let us say we have a disease, and for each patient, the state of the disease is unobserved, and we model it with the binary random variable $X:\{\text{patients}\}\longrightarrow \{0,1\}$, where $X = 0$ when the patient is health, and $X = 1$ when the patient is infected.
We have a diagnostic method, but we don't know if it can absolutely give the correct result. So, we model it using another random variable (but it is observed), $Z:\{\text{patients}\}\longrightarrow \{0,1\}$, where $Z = 0$ when the diagnosis is negative and $Z = 1$ when the diagnosis is positive.
We don't know how to model $P(X=0)$ nor $P(Z=0)$, but we know the conditional probability $$P(Z = z|X = x),\ \text{where}\ z,x\in\{0,1\}.$$
For example, we can then know the probability of the diagnostic method missing the presence of the disease, where is: $$P(Z=0|X=1)$$
However, is there any way to get he probability of the diagnostic method gives the correct result from the condition probability? I am not sure if the following is correct: $$P(\text{the diagnostic method gives the correct result})=P(Z=0|X=0)+P(Z=1|X=1).$$ I don't think that it is correct, but I don't know what the correct alternative is.
By definition of conditional expectation $P(A \mid B) P(B) = P(A \cap B)$ (assuming $P(B)$ is positive). Note that the event you are interested in is $\{X = Z\}.$ Then $$ P(X = Z) = \sum_{t = 0,1} P(Z = t \mid X = t) P(X = t). $$