I once flipped a coin. It landed on its side. #lolvariance Improbable Things Happen.

Vince had a pretty clear 50,000 hand downswing either side of the 200k mark. /pedant mode
You've read Verneer's book, so you've seen the original graph, and should understand that there were several downswings that don't show up clearly in the graph for the long run. In poker, the "long run" is very long indeed. That's kind of why I hate analysing results from 10k hand samples. They are virtually meaningless.

That's a very impressive graph!

Both random samples of 10k coinflips.

Lots of good replies about the actual math. Here are a few other ideas.

People tend to filter the world through their expectations. Notice when you talk about luck here, you only mention downswings. Why is that? Nearly every poker player has a filter on their perception of results

• Good results = skill
• Bad results = bad luck
• Less often --> bad results due to single problem "which I've now fixed"

Inevitably you have some new LHE poster who ran 3BB/hr for 50 hours in his local 4/8 game or some NL poster who ran 25bb/100 at 10NL post a brag that amounts to "I've solved poker" explaining his win as being a much better (something) than his opponents, admitting he's running a little hot, and explaining why he probably is a 15bb/100 winner long term. Even the guy who has clearly been running super hot biases his results as mostly skill. The typical near breakeven to slightly losing new player who runs hot to win at the clip of a solid pro, he just feels like his standard win rate is normal. All this to say that people overestimate their results. Why all the downswings? People are dishonest with themselves about actual expectation.

Also people compute downers by looking at their graphs, finding a peak, and then any $ lost below that peak is a downswing. Basically, they maximize their measurement of downswings. Every poker player who looks at results this way is maximizing his pain. I know a guy who crushes live games in LA as a pro, according to this measurement he spends 80% of his time on a downswing. Think this through. Do you see why this actually produces maximum pain?

Even the EV corrected graphs are BS in many ways. Here's a quick example. Let's say I overplay QQ and JJ, so that when I get them I only get AI vs KK+. Mostly, I'm winning small pots as bluffs and once in a while I lose whole stacks in coolers. I look at my luck long term and have slightly fewer suckouts on KK+ than standard, so I'm running really bad and losing money with QQ and JJ, right? No, I'm butchering those profitable hands and making them unprofitable. AIEV gives me an incorrect perception.



Poker is swingy. Accept that. Look at the quality of decisions you're making, rather than results. Try to get here
Also, know that you're biased to give yourself the benefit of the doubt (explain upswings as skill, downers as luck) and to use a terrible measuring stick for downswings (your most recent peak). Realize that as a biased observer, your opinion is likely wrong.

No, that's not why. Yes, the Martingale system is a surefire path to ruin, but not because of variance, or losing streaks, or anything along those lines. The real reason the Martingale system doesn't work is that no one has infinite money to spend (to cover a colossal losing streak which will eventually happen if you play long enough), and probably more importantly casinos have limits on how much you can bet. Eventually you're going to want to double your bet and the casino will say "Uh.... no." Then you're screwed.

To be fair if the variance of Martingaling on every single bet was 0, then it would be clearly +EV, no?

Depends what game you're talking about. If it's a -EV game then no, without variance you'd lose (edge)*(wager) on every bet. If it's a +EV game then sure, martingaling is +EV and +EG (expected growth) in the absence of variance (though, better would be to always wager your entire bankroll).

One thing that doesn't get mentioned but should: with variance, the martingale is so bad that you go broke even if making +EV wagers. That's because martingaling is -EG. In a finite session it's +EV while being -EG, but the EV of playing for eternity is negative because your chance of going broke becomes 100% (which is implied by the EG being negative). That's how bad the Martingale is.

I have 2 streak programs on my calculator. One only works for p=.5 (it uses generalized fibonacci) and that's the one that had the memory overflow. The other one works for any p but is slower since it has to evaluate a summation. However, just now I optimized the code (to my ability) and didn't have to wait very long for the answer: P(streak of 10+ tails in 1000 flips) =~ 38.545%

For those who have a TI calculator and want the code:

I admit I don't understand the question. How can the variance on a single bet be 0? The definition of variance requires there to be multiple bets.

Unless you're using unnecessary redundancy. (Yes, I understand the irony of that, but hey - fight fire with fire! ) In other words, "Martingaling" implies you're applying it to all bets. The system makes no sense if you don't.

But still, I don't understand that question either. The whole premise of Martingale is to deal with the variance. The only way the variance could actually be zero is if there was no randomness. For example, the results of coin flips were guaranteed to be HTHTHTHTHTHTHTHT. If so, then what's the point of this conversation to begin with?

Maybe this will answer your question, maybe it won't. There is "Martingale in theory", and there is "Martingale in practice, in the real world."

The concept of Martingale is appealing, because in theory it works. But the theory requires concepts that don't exist in the real world - specifically, the concept of infinity. So Martingale does work in theory, but those conditions don't exist in the real world, and therefore Martingale does not work in the real world. There is no such thing as "an infinite amount of money" or "an infinite amount of time".

And that is why Martingale fails. If you play long enough, then you will eventually hit a losing streak so monstrous that it will blow away any small gains you've previously made, and ultimately leaving you with no way to recover. Once your bet is as big as all the money in the world, what will you do? Or more to the point, once your bet is bigger than what a casino will accept, what will you do?

The practical reasons it fails are much more modest. A Martingale system is designed to maximize the chance to win one minimim bet. One. So you win $10 if that is the unit bet. But if you play 1000 spins on a $10 American roulette table, there is about a 1% chance you'll have to put over a million bucks on the table just for that chance to win $10 (if the house lets you).

Nobody is degen enough to smack down a mil to win $10. At some point everyone either gives up or goes broke with Martingale. Or they win $10 and quit immediately and never bet again. That's the only way to win with it.

Put up the moon as collateral, ldo.

Is that the chance to hit a streak of 10 tails specifically, or the chance for a streak of either 10 tails OR of 10 heads?

I'm pretty sure it's the latter.

Heads specifically.

And 38.5% sounds about right to me. 10 in a row in 10 flips is 1/1024. We're talking about 1000 flips. To have 1024 non-independent sets of 10 would require 1033 flips. 1024 independent sets would give a probability of almost 1-(1/e). Non-independent sets result in a smaller chance, but I wouldn't expect it to drop from 63% to 10% just because the sets aren't independent.

I'm not sure one would calculate the "either/or" chance. You can't just multiply 38.5% by 2 because you'd be over-counting the chance of both. I guess you can do that and subtract the chance of both, but the hard part is finding the chance of both. I may have seen the solution once but have forgotten it. I think Bruce' streak calculator in Excel can do it.

Oh come on - we've all been there!

Would if I could!