You can't prove it's impossible!

[reddragdiva]

A common sophistry which really annoys me is the one that conflates an utterly negligible probability with a non-negligible one. The argument goes:

1. There is technically no such thing as certainty.
2. Therefore, [argument I don't like] is not absolutely certain.
3. Therefore, the uncertainty in [argument I don't like] is non-negligible.

Step 3 is the tricky one. Humans are, in general, really bad at feeling the difference between epsilon uncertainty and sufficient uncertainty to be worth taking notice of — they can't tell a nonzero chance from one that's worth paying attention to ever. (This is why people buy lottery tickets.)

It’s a terrible, terrible argument, and an unfortunately common one. It needs to be bludgeoned to death every time it’s brought up.

Comments

[doug]

"can't tell a nonzero chance from one that's worth paying attention to ever. (This is why people buy lottery tickets.)"

Good point ... that isn't enhanced by choice of final throwaway example, I reckon.

You're skating close to a fallacy yourself there. Let's start with saying that a one in fourteen million chance (main game in the UK National Lottery) is effectively the same as no chance at all. (Which is almost what you were saying.) Let's imagine a slightly differently structured lottery to most, with fourteen million tickets numbered sequentially, all sold, and one of which is then drawn at random. Ticket number one has no chance. Ticket number two has no chance ... ticket number fourteen million has no chance. But one of them will win.

There can be a really significant difference between epsilon probability and no probability; and there can be a really significant difference between epsilon and a merely very unlikely probability like one in fourteen million chance.

Lotteries are (usually) a really bad example of low-probability events, because we can be extremely accurate with our estimates of how probable they are. What's the chance of you winning the lottery? Treating it as pretty likely is wrong. Treating it as zero is also wrong. Treating it as epsilon is also wrong. It should be treated it as 1 in 14 million (to 2 sig fig).

(... you've almost triggered my rant about how it can be entirely rational to play the lottery, but this margin of my time is too small to contain it. Hint to part of it: People's utility function is not linear with money. If I find time I'll post it on my own journal and post a link here.)

[reddragdiva]

Treating it as one in fourteen million is, I suspect, perceptually impossible for humans without training. I suspect that without practice people don't have a feel for less than ten percent. (I sure don't and I've actually tried to, therefore everyone else has the same problem ... but they do in fact seem to.) But people observably just don't shut up and calculate even as they concede the lottery is way below their perceptual level.

I wonder if work's been done to measure just how good humans are at feeling probability, with or without practice.

[reddragdiva]

No, people's utility function with money appears to be logarithmic. (No link to hand, I think I can find one if needed.)