Why an error probability of 1/3 in BPP? BPP is defined as the class of polynomial-time random algorithms which have an error probability of at most 1/3.

But why was 1/3 chosen? If we have an algorithm with some error probability less than 1/2, then we can run it several times, taking the most common result, to obtain an error probability of less than 1/3 while still staying in the same complexity class.

So why isn't BPP instead defined as the algorithms which have an error probability of less than 1/2? Is there something special about 1/3?
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.