theferrett just claimed that he'd written a unique sentence, and was a tiny bit proud:

Trillions of English words are spoken daily, in many configurations. At this point, it's hard to put together a reasonable sentence that hasn't already been said by someone else; sure, you can mash together a set of random words like "May I mambo dogface in the banana patch?", but for most of the things you'd want to convey you have to realize that you're trodding a well-worn path.

Every once in a while, though, you get the pleasure of speaking a sentence that you can be almost certain has not been spoken before by any human, living or dead. And then you sit back and such in a deep, happy sigh at making a little notch in the world - even if no one ever sees it, you know that you have been as unique as it is possible for an organism to be.

My contribution?

"I invited my periodontist to play Rock Band: Beatles."

Obviously an amusing throwaway post, but it occurred to me that actually pretty much any sentence over about 9 or 10 words was unique because of how many possible sentences there were. I pointed that out, and he wasn't impressed. I thought I'd reproduce (and lightly refine) the math here, because it's really mind boggling, and really counterintuitive until you see it.

So let's say you have roughly 40,000 words in your vocabulary, counting brand names and proper nouns. That puts the cap on the number of 10 word sentences you could generate at 10^26 assuming no grammar rules. Grammar constrains the hell out of that, so let's say only one in a million of those is a real sentence that someone might say. So we have 10^20 possible sentences.

Ok, on the other side, how many 10 word sentences get spoken or written in a day. Let's say 10 per person, because most of your spoken sentences are short, and the number of words in a sentence is super variable (this sentence has 38, for instance), so 10 words is a pretty small target. But it could be 100 and wouldn't make much difference.

How many english speakers are there in the world? Let's say a billion, which I'm sure is an overestimate of the number that speak english day to day. So that's 10 billion candidates per day, or trillion candidates per year. That's 4*10^12, or to put it another way, it'll take like 10,000,000 years for all the 10 word sentences to be uttered assuming no overlap.

Let's run the calculation backward. English has been around for less than 1,000 years, so let's use that as a starting point. What's the average number of speakers over that time? Can't possibly be more than 100 million, so 10^8 people times 10^3 years times 10^4 10 word sentences per year per person (again an overestimate) and we get 10^15 10 word english sentences uttered throughout history. So for any given 10 word sentence, you have a one in one hundred thousand chance of uttering a previously spoken sentence assuming the language hasn't changed at all in the last thousand years.

The odds only get worse as the number of words goes up. Each extra word adds at least a couple of orders of magnitude to the number of candidate sentences, and maybe quite a bit more. So revel in the fact that every longish sentence you say is unique, yours, and nobody else's.

Last week I got all in with QQ versus AA to lose a big pot. I decided that it was a pretty clear fold instead of a call, but now I'm not so sure. I still think it's a good exploitative fold, but folding that there against everyone leaves me open to being exploited, I think.

Let's construct a similar situation that slightly simpler and look at the math. Villain limps for 40 with a wide variety of hands, as before -- let's say his range is any pair, any suited A, lots of suited hands, and any two broadway cards, which is about 30% of the possible hands. I think his range is something like this, though I think he probably raises with his better hands before limp-reraising with his best ones, but let's just leave them all in for now.

He gets raised to 220 by someone trying to isolate, and I raise to 700 behind. I think the isolation raise is not very tight, so I'm raising for value with AQ+/99+, let's say. We'll stipulate that I have 3000 total, so 2300 back, and that the limper will now jam or fold., and that the other raiser will fold.

If I just call with AA or KK, he should jam every hand, I think. I'd be folding 56 hands and calling 12 (since there are 16 ways to make AK or AQ and 6 ways to make each pair). His EV is $1000 when I fold, and when I call he wins 20% of the time and loses 80% of the time, so a net -$1740. He breaks even if I fold 64% of my hands, since $1000*.64 - $1740*.36 is +$13.6. But I'm actually folding 85% of my hands here if I fold QQ!

In order not to be exploited there, I need to call 24 hands, which is either JJ+ or QQ+ and AKs and a couple times with AK. Now he does slightly better again because he has 25% equity instead of 20% equity when I call, but it's not a big difference.

Ok, let's take a step back. I can't fold too often because I leave myself open to being exploited. He has to worry about the other direction, though -- if he does jam with everything, I can simply call with everything and crush him because my range is so much tighter. What should his jamming range be there? He's putting in $3000 to win a $1000 pot if I fold, and a $3300 pot if I call. Conversely, I need to call $2300 to win $4000, so I need to win 36% of the time to call.

So let's assume I call perfectly correctly against his range when he jams. What should he jam with? Hm, I don't actually know how to calculate that. Well, let's play around with pokerstove and see if I can find an approximation.

If he jams with JJ+/AK, I call with... actually, exactly the same range. TT isn't good enough (33% equity), AQ is obviously awful, JJ is just good enough and AK is plenty good. So I call with 60%ish of my hands, which means he wins .4*1000 + .6*150 which is a nice profit. Hm, ok, let's lock in my calling range at JJ+/AK. His EV is (.4*1000 = .6(6300p-3000) where p is his win percentage against my range. So solving for p gets me .37, so he should jam with any hand that is .37 or better against my range. So JJ is very borderline, but QQ+/AK are good.

Should we just iterate? Now I respond by calling with my exact same range. So we seem to have converged.

Ok, so unless I did something wrong up there, which is likely, it looks like he should jam in that situation with QQ+/AK and I should call with JJ+/AK. He makes $740 with this play.

So now let's unroll some of those assumptions. In particular, there were two questionable assumptions I made: first, that he limps with all his hands, and second, that the other person never calls. I think both of those things conspire to make our limper-reraiser tighter, though I'm not sure how much. I think he opens for more than a limp for at least some of the time with QQ or AK; second, I think he has to be a little careful of the other guy who has a pretty big stack and could have a hand here. So if we reduce his range to QQ+/AKs, for instance, now JJ is no longer a calling hand for me, and neither is QQ.

Actually I suppose I could calculate it. The other player has like broadway cards/66+/suited connectors/some suited aces, say. He's getting better odds, but he has to beat two players, and people who are not me have 7000 back after my 3k, so he has to call very tight here. This is getting long so I'm going to say he jams with KK+ only, which is 5% of his hands. So for me it's very simple: if he comes, I fold anything but rockets. The limper also has to fold KK, so his EV now looks like, assuming his earlier range of QQ+/AK, and when the guy jams:

.05(.18[.77*10800-.27*10000]) = $73, so basically this is negligible, since it costs him less than $100 in EV but he was making $740 with his earlier play. I'm ignoring some card deletion effects here but I don't think they matter much.

So we come to this: if the limper never opens for more than the minimum, given the action he should raise to $3000 with QQ+/AK and I should call with JJ+/AK, though I could fold JJ too, since it's basically a wash for me. But the point is that I should call with QQ in that situation against an optimal player who knows my raising range. However, if the player would open for more with QQ/AK hands so I can reduce them in his distribution, then I should fold QQ there, and AK as well. That seems likely to me, so I probably should have folded my QQ after all.

Hm. This is slightly puzzling to me. Way earlier, I noticed that calling with AA/KK only would leave me open to exploitation. Now it seems like that's the right play... but it still leaves me open to exploitation. Since an optimal strategy can't be exploited, I must have screwed up my reasoning somewhere. Any thoughts out there?