# Appeal to probability

It's possible that they cause good luck; as luck would have it though, it's damn unlikely. (Source)
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An appeal to probability argues that, because something probably will happen, it is certain to happen.

Alternate forms include the appeal to possibility (it is possible, therefore it is certain) and the appeal to improbability (it is improbable, therefore it is impossible).

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

The fallaciousness of this line of logic should be apparent from the second, unstated premise (P2), which seems and is blatantly false.

### Appeal to possibility

This form argues that, merely because something is possible, it is certain to happen.

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

### Appeal to improbability

Conversely, the appeal to improbability argues that, merely because something is improbable, it is impossible.

P1: X is improbable.
P2: (Unstated) Anything which is improbable, is impossible.
C: X is impossible.

## Explanation

The appeal is based on a faulty premise — that probability/possibility are the same as certainty, or that improbability is the same as impossibility. There is no support for this premise, anywhere.

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. And even though it's probable that you will lose, it is not certain that you will lose. Even if there was a 99.9999% chance (highly probable) that you would win (you bought 999,999 tickets), it is still possible that you might lose (to your previous self, with your previous one ticket) — so it is not certain that you will win.

Another way of looking at this is through game theory. Game theory states that the value of something is equal to its probability times its worth. For the lottery example above, 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 probability, then of course! According to the appeal to probability, if there's a 0.0001% chance of winning, that's the same as a 100% chance of winning. 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 — if you played this lottery an infinite number of times, you would on average win only 0.0001% of the time.

## 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]