What does proof on a balance of probabilities mean?
The civil standard of proof is proof on a balance of probabilities. The criminal standard is proof beyond a reasonable doubt. This article deals with civil standards of proof. 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%. So mathematically proof on a balance of probabilities is 50.1% likelihood of something having occurred.
A lawyer I once knew explained it this way. You see the scales of justice. They are evenly balanced. Both sides are the same height. But if one side has the weight of a feather added to it, causing it to go down and the other side to go up. Now that side with the weight of a feather has won. Proof greater than that is not required. Simply a balancing of both sides seeing which side has the stronger proof.
Quotes from cases where judges have commented on the civil standard of proof follow:
McIver v. Power,  P.E.I.J. No. 4, Prince Edward Island Supreme Court – Trial Division, MacDonald C.J.T.D.,
5 In any civil case the plaintiff must prove their case on a balance of probabilities if they are to succeed. This means that the plaintiff must prove that his facts tip the scale in his favor even if it is only a 51% probability that he is correct.
F.H. v. McDougall,  S.C.J. No. 54:
44 Put another way, it would seem incongruous for a judge to conclude that it was more likely than not that
an event occurred, but not sufficiently likely to some unspecified standard and therefore that it did not occur.
As Lord Hoffman explained in In re B at para. 2:
If a legal rule requires a fact to be proved (a “fact in issue”), a judge or jury must
decide whether or not it happened. There is no room for a finding that it might have
happened. The law operates a binary system in which the only values are zero and one.
The fact either happened or it did not. If the tribunal is left in doubt, the doubt is resolved
by a rule that one party or the other carries the burden of proof. If the party who bears
the burden of proof fails to discharge it, a value of zero is returned and the fact is treated
as not having happened. If he does discharge it, a value of one is returned and the fact
is treated as having happened.
In my view, the only practical way in which to reach a factual conclusion in a civil case is to decide whether it
is more likely than not that the event occurred.
 In the result, I would reaffirm that in civil cases there is only one standard of proof and that is proof on a balance of probabilities.
Snell v. Farrell,  2 S.C.R. 311 is a medical malpractice case and the issue was whether the plaintiff required a firm medical opinion to establish a causal link between her surgical outcome and the defendant doctor’s actions. Sopinka J., speaking for the Supreme Court of Canada, commented on the difference between medical certainty and legal certainty. He commented that “near certainty” is the medical standard of causation while the legal standard requires only a 51% probability. He concluded that a firm medical opinion is not required to establish causation in law. Rather, causation is a question to be determined by weighing all of the evidence in order to determine whether a causal link has been established on a balance of probabilities.
Veerasingam v. Canada (Citizenship and Immigration), 2012 FC 241 — Federal Court of Canada
. . .paras 29 that the term “balance of probabilities” was equivalent to “more likely than not”, but with two distinct steps involved as to the burden of proof and legal test. […] to be persuasive evidence of that risk “on a balance of probabilities” or to show that it was “more likely than not.” The Board uses this terminology throughout its decision. […] As demonstrated, the Board was mindful of its role in considering the evidence of “risk” to the Applicant on a balance of probabilities. . .
Roseanne Conley v. Keel Construction, 2005 NBQB 263
 On the question of liability the onus of proof must be met by the claimant, Mrs. Conley and the standard of proof is that of a balance of probabilities, i.e. is it more likely than not that the incident occurred in the manner she alleges.
Bhullar v. I.C.B.C., 2009 BCPC 44
But the burden of proof here is not like a criminal charge, beyond a reasonable doubt. The burden of proof is on the balance of probabilities, more likely than not. . . While I have an expert opinion, which I accept, that this car was either driven with a key or towed, for the reasons I have already gone into, I do not find this vehicle was towed, which means the likelihood is, more likely than not, balance of probabilities proven the vehicle was driven with a key. The only people that had keys were the claimants. Nobody else. If that is true, which seems to be true on the facts, then again more likely than not, Mr. Bhullar was involved in the theft and burning of the vehicle.
Mitcham v. Canada, 2003 FCA 340 Federal Court of Appeal
. . . While Mr. Mitcham is now diagnosed as having fibromyalgia symptoms, the Board, weighing the evidence on a balance probabilities, held that he was more likely than not capable of regularly . . .
R. v. Oakes,  1 SCR 103 — 1986-02-28 Supreme Court of Canada
. . . in such a case and it is sufficient if the court considers that upon the evidence before it it is more likely than not that the fact does not exist. . . . The test is the same as that applied in civil proceedings: the balance of probabilities. . .
Canada (Minister of National Revenue) c. Fabrication GMCA Inc., 2002 FCT 1260 Federal Court of Canada
. . . (3) The evidence must show, on a balance of probability, that it is more likely than not that collection would be jeopardized by delay. . . Lamer J. for her to conclude that on a balance of probabilities it was more likely than not that giving the respondent further time could compromise the collection in question. . .
R. v. Turner, 2004 BCSC 1778, Supreme Court of British Columbia
. . .  The standard of proof in civil trials is proof on a balance of probabilities; namely, proof that something is probable or more likely than not. . . . . In criminal trials, such as this trial, the standard of proof beyond a reasonable doubt is higher than the civil standard of proof on a balance of probabilities. . .
So you can see the test of proof in a civil case is whether it is more likely than not, that something occurred. No more, no less.