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?