Why an error probability of 1/3 in BPP?
The constant $1/3$ is completely arbitrary. Let's say that $\mathsf{BPP}_p$ is the class of problems solvable in polytime with error at most $p$. Then $\mathsf{BPP}_p = \mathsf{BPP}_q$ for any $0 &lt; p,q &lt; 1/2$. The simple proof can be found in any decent textbook or lectures notes on complexity theory, and is also a nice exercise. The idea is that you can boost the success probability by running the algorithm several times and take a majority vote (details left to you).

Why do we like $1/3$, then? It's the "simplest" rational number in $(0,1/2)$: for example, it is the rational number in $(0,1/2)$ with the minimal possible denominator, and with the minimal sum of numerator and denominator.