Let the correct answer be $\xi$ and the outcome of the repetitions be $x_1,x_2,\ldots,x_n$ respectively. $\Pr{(x_i=\xi)} = p \quad \forall i$ This makes the overall experiment a binomial distribution with $n$ attempts and $p$ probability of succeeding in each trial.
If $\xi$ is the median of the $n$ trials, then the necessary and sufficient conditions are $\Pr{(x_i\geq\xi)} = 0.5 \quad \& \quad\Pr{(x_i\leq\xi)} = 0.5 $
The probability of this happening is the answer you are looking for i.e. $\Pr(\Pr{(x_i\geq\xi)} = 0.5)$. In order to do this, you need to know the skewness of your errors i.e. what are the chances that the output will be greater than the real answer . The data given is insufficient.
A bound can be given by the observation that if out of $n$ observations, if more than half of the values are equal, then they are also equal to the median. So, the lower bound on this will be given by the probability that at least half of the answers are correct. It can be calculated straightforward by using CDF of binomial distribution with parameters $(n,p)$