# Appeal to probability

It's possible that they cause good luck; as luck would have it though, it's damn unlikely. (Source)
 Cogito ergo sumLogic and rhetoric Key articles General logic Bad logic v - t - e

An appeal to probability, also known as appeal to possibility, is the logical fallacy of assuming that something will happen simply because it is possible that it will happen.

Alternate forms include the appeal to probability, where the occurrence of an event is assumed even though no evidence is provided for the event, but instead fallacious reasoning is used in order to overstate the probability of the event.

The fallacy is an informal fallacy.

## Form

The argument, when expressed, usually takes the form:

P: X is possible.
C: X is certain.

However, the argument relies on the form:

P1: X is possible
P2: (Unstated) Anything which is possible, is certain.
C: X is certain.

The fallaciousness of this line of logic should be apparent from the second, unstated premise (P2), which is logically inconsistent.

## Explanation

The appeal is based on a faulty premise — that possibility is the same as certainty, or that uncertainty is the same as impossibility. This premise is not supported and logically inconsistent.

Consider a lottery with 1,000,000 players and 1 winner. If you bought one ticket, it is possible (at 0.0001%, the chance is nonzero) that you might win the lottery; however, it does not follow that you will win the lottery. Let's say the lottery pays out \$5,000,000 (a tempting sum) and costs \$10 (a trifle). Is it worth it? If we accept the appeal to possibility, then of course! According to the appeal to possibility, we ignore the improbability of winning and just assume that we will win. Thus we should play because we want the \$5'000'000. Yet this is not borne out by the evidence. Your realistic expectations of gain are laid out in the table below:

Probability: Worth: Value: 0.0001% +\$5,000,000 0.000001 x +\$5,000,000 =  +\$5 100% -\$10 1 x -\$10 = -\$10 -\$5

Buying a ticket, on average, loses you \$5.

The law of large numbers supports this — as the number of games you play diverges to infinity, you would on average win only 0.0001% of the time almost surely.

Note also that, by construction of the lottery, there will be exactly one winner. Therefore, the statement "it is very unlikely that a randomly chosen player wins the lottery" is true, while the statement "it is impossible that any player wins the lottery" is false.

## Examples

An example of the appeal is Murphy's Law — if something can go wrong, it will. (This is more an example of unending pessimism rather than a logical argument, so it doesn't quite count).

Other examples:

• There are so many religions so one of them has to be correct.[1]
• There can be many worldviews — and all of them wrong!
• I will never reach this goal because it is possible that I cannot reach this goal.[2]
• I'm certain to win the lottery if I just keep buying tickets.[3]
Simply, you may never win the lottery. And as noted above, it costs more money to purchase the ticket than you can expect to get back, so it's not even worth trying.
• There are so many stars in the Universe that it is certain, that not only is there intelligent life out there (see the Drake equation), but that it has visited the Earth.[2]
• There are many hackers on the internet, therefore, you will be hacked.[2]
• I’m going to play professional basketball when I grow up. I don't need to worry about my grades much since I'll be making millions after I am drafted into the NBA. I know there are only a couple hundred professional slots, and millions of aspiring young players, but since I have as much a chance as any other young player, I will succeed.[3]

### Appeal to possibility

From Logically Fallacious:[4]

Dave: Did you know that Jesus was gay?
Tim: And why do you say that?
Dave: You have to admit, it is possible!
Tim: So is the fact that you are a moron.

## Exceptions

### Science

The first and major exceptions are that of science. Science bases all of its evidence on probability — by showing that alternate hypotheses are extremely unlikely, science shows that its favored hypothesis is extremely likely. Science is exempt from the appeal to probability for exactly that reason — science never asserts that something is certain, merely extremely likely. The difference between a "fact" (something certain) and a "scientific fact" (something extremely likely) should make this apparent.

### Certain or impossible things

The second major set of exceptions is when something truly is either certain or impossible. If something has a 100% chance (a certainty) of being true then necessarily it will happen. The opposite is true for something that has a 0% chance (an impossibility) of being true. These cases are usually only found in formal logic and mathematical proofs, where a set of axioms are assumed to be 100% true, for the purpose of argument. So, in truth, these results are "certain" or "impossible" only within the bounds of the axioms that support them.

### Large risks

The third major set of exceptions are risks which, if they ever came true, would pose such a large threat that they should be avoided at all costs (or at least, have their probability reduced as far as possible) — even if there's only a chance that they occur.

Essentially, one must use game theory again: even a very small probability of a very bad event is sufficient to make it very important.

One example is a concussion: repeatedly ramming your head into a wall isn't certain to give you a concussion, but it's not worth the risk (the benefits don't outweigh the harms). Another is nuclear war: because it could possibly (though this is unlikely) kill everyone on the planet, it has been argued that no other goal takes higher priority than reducing the risk of nuclear warfare. Unfortunately, our politicians have not taken the same attitude to anthropogenic climate change and other environmental degradation, even though it is far more likely to cause the collapse of modern civilization than nuclear war ever was.

One counterexample is that of a small meteor strike: simply put, the real (nonzero) possibility that a small meteor will strike your house while you sleep does not warrant your concern, because [1] there is nothing you can do to stop said impact and [2] the impact of such an event would be relatively small.[3]