Why is the probability used in the definition of RP complexity classes, arbitrary?
Yes, the constant is entirely arbitrary. Call an algorithm a $p$-algorithm if:


When the correct answer is NO, it always answers NO.
When the correct answer is YES, it answers YES with probability at least $p$.


Note that the probability in the YES case is only over the algorithm's coin tosses. 

Given a $p$-algorithm and a constant parameter $m$, we construct a new algorithm which runs the original algorithm $m$ times (with independent coin tosses) and takes the OR of the result. This algorithm is a $1-(1-p)^m$-algorithm (exercise). For every constant $q &lt; 1$ you can find a constant $m$ such that $1-(1-p)^m \geq q$. Hence if a problem admits a $p$-algorithm it also admits a $q$-algorithm, for every $0 &lt; p,q &lt; 1$.