Saying something is proven on a balance of probabilities means that it is more likely than not to have occurred. It means that it is probable, i.e., the probability that some event happens is more than 50%. On the otherhand, if a legal case or a person's guilt is proved beyond a reasonable doubt, there is enough proof for the person accused of a crime to be judged guilty. i.e., it's around 100% likely that the events have occurred.