Error Rate comparison between a Random Guess and a Bayesian Guess

For a 4-class problem and a known feature vector $\mathbf{x}=\mathbf{O}$ you obtain the following posteriors:

s	1	2	3	4
P(s | x=0)	0.1	0.15	0.7	0.05

What is the error rate, if you randomly throw the dice? What is the error rate, if you employ the Bayes classifier?

• Random

$$\begin{array}{rl}\text{Error Rate}& =1-\frac{1}{n}\\ & =0.75\end{array}$$

• Bayes Classifier Chooses the class with the most probable A Posteriori.

$$\begin{array}{rl}\text{Error Rate}& =1-\arg \underset{i\in \mathcal{S}}{\max }p(s=i\mid \mathbf{x})\\ & =1-0.7\\ & =0.3\end{array}$$