True Loss for Bayes Classifier with Two Classes

Question

For a Bayes classifier of two classes (say 0 and 1), I'm not understanding how the largest possible true risk would be 0.5? I'm assuming that we assign a 0 loss for a correct classification and a loss of 1 for a misclassification. So does that mean that if the probability of the posterior $p(y=1|x)$ is less than half, then the Bayes classifier picks a 0 and then 1 otherwise. Then there's 0.5 chance that $p(y=1|x) > p(y=0|x)$ and a 0.5 chance that $p(y=1|x) < p(y=0|x)$. And if the actual label is a 1, there's a $(0.5)(1) + (0.5)(0)$ classification loss; and then if the actual label is a 0, there's a $(0.5)(0) + (0.5)(1)$ classification loss? Not sure if this is how it's done, when calculating largest possible true error rate. Does the calculation method also work for say 3 classes? Thanks in advance!