Math question with no real world significance

[Zetack]

In another forum a poster made a somewhat trollish statement that every poker player would go broke "eventually".

The reasoning being that, no matter how small your risk of ruin, given enough trials (an infinite number, if necessary) you will always hit your percentage chance of ruin, no matter how small.

Well that thought didn't interest me particularly, but it got me to thinking about math, and I since I suck at math, its very possible that I've broken my thinker and my head is now spewing out gobbeldygook. So let me run by my line of thinking, with an actual question at the end.

So I started with the idea that an improbable event becomes probable with enough trials. I.E. a million to one shot is pretty likely to occur if you run a hundred million trials.

so in relation to poker, if the deck is completely random, and if your odds of getting a specific pair, say 99, are 220-1. Then you odds of getting that specific pair back to back would be, what, 48,400-1? Pretty long, but if you play enough poker you're going to see it from time to time (heck I was dealt KK back to back in a live tournament on Sunday.)

And then your odds of getting your 99 three times in a row would be somewhere in the vacinity of 10 million to one. Ok, I'm not likely to see that, put you could envision poker playing computers getting thousands of hands per second hitting these odds.

So if you keep running the streak further out, pocket nines ten times in a row, 20, 1000... no matter how low the probability, if you ran enough trials eveuntually you would get a streak that long. Right?

And if you have an infininte number of trials, it doesn't matter how long or unlikely the streak is, eventually you will hit it. Given an infinite number of trials eventually you would have a streak of say pocket nines a million times in a row.

And it doesn't matter how big the number. A million times in a row, a billion, a quadrillion, evenutally you will hit the streak.

Now assuming the above is correct (is it, anybody?) Then I started to wonder, and here's where my head starts to hurt, given an infinite number of trials, will you eventually get a streak of pocket nines that itself goes on for an infinite number of draws?