What it means
The balance of probabilities is the standard of proof in civil cases in Australia. A party succeeds on an issue if the court is satisfied it is more probable than not, often described as more than a 50 per cent likelihood. It is a lower standard than the criminal "beyond reasonable doubt". For very serious allegations, courts will still require clear and convincing evidence before being satisfied on the balance of probabilities.
How it's used
The judge accepted, on the balance of probabilities, that the contract had been signed by both parties.