The Gambler's Fallacy

I have been a serious part-time counter for several years. I have given up counting for a while (for various reasons) and have been trying out a system that I have put a lot of thought into. I have been playing it for about 5 months---have been very successful---but I know with the few # of hands played (just over 11k) there is no reason to get excited. Here is my system:

I keep track of the number of hands lost vs. number won. I know, in the long run, the win/loss % is 47.5/52.5. I count on long losing streaks and then the math to take over and start bringing me back towards that 50/50 ratio. For example:

I start out betting 2 units. I count my wins and losses. Win = +1, Loss = -1. I bet 2 units until I reach a -6 count. At that point I ramp to 10 units. I continue to play untilt the count is even (I stop), or the count goes to -12 (I ramp up to 15 units. I continue to play until the count is -2 (I stop) or the count goes to -18 (I ramp up to 20 units). I continue to play until the count
reaches -5 (I stop) of goes to -25 (I ramp up to 25 units).

I have never gone beyond -28, and this only one time in 5 months.

I know this seems ridiculous to some of you. However, just as sure as the math says that the casino has a small edge over perfect basice strategy, it also says that my long term win loss ratio will approach 48/52.

I like to play 100 hand sessions. Occasionally, if the count is high, I will go up to 200 hands. Yes, I have losing sessions (about 1 in 6), but until now my wins have far exceeded my losses.

My heart wants to believe this will work long term, but my head tells me this has already been tried and there is a professional out there that can show me the error of my ways! I have learned a lot about blackjack and counting from this website and I would like to thank the pros who take the time to respond to questions like theses!

adventureboy said:

I count on long losing streaks and then the math to take over and start bringing me back towards that 50/50 ratio.

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Your system is based on the Gambler’s Fallacy:

“If a fair coin is tossed repeatedly and tails comes up many times in a row, a gambler may believe, incorrectly, that heads is more likely on the following toss. This is an informal fallacy.”

http://en.wikipedia.org/wiki/Gambler's_fallacy

There is no reason to think that the wins and losses are going to immediately start evening out. In fact, it may take thousands or even millions of hands for the number to approach the expected ratio. What makes things worse is that even though the percentages are getting closer to what you would expect, the absolute differences will tend to get bigger!

For example, you would expect a fair coin flip to have a 50/50 distribution of heads and tails. But what happens when you bet your money on it? Let’s see.

You decide to bet $1 on tails and see how things go. After 10 flips there have been 8 heads and 2 tails. You have lost $6 so far. Using your logic you would continue betting on tails because you expect it to even out to 50/50. After 100 flips there have been 55 heads and 45 tails. That’s a little closer to 50/50, but now you’ve lost $10 in the process. After 1,000 flips there have been 515 heads and 485 tails. Still closer, but now you’re down $30. After a grueling 10,000 flips there have been 5050 heads and 4950 tails. Now we’ve got a 50.5/49.5 ratio, which is darn close to 50/50, but you’ve lost $100. Even though the percentages are getting closer to 50/50, the actual number of flips is getting farther away and you’re losing money along the way.

Listen to your head and ignore your heart. I know that you want this to work, but I can tell that you know better.

-Sonny-

Sonny said:

Your system is based on the Gambler’s Fallacy:

“If a fair coin is tossed repeatedly and tails comes up many times in a row, a gambler may believe, incorrectly, that heads is more likely on the following toss. This is an informal fallacy.”

http://en.wikipedia.org/wiki/Gambler's_fallacy

There is no reason to think that the wins and losses are going to immediately start evening out. In fact, it may take thousands or even millions of hands for the number to approach the expected ratio. What makes things worse is that even though the percentages are getting closer to what you would expect, the absolute differences will tend to get bigger!

For example, you would expect a fair coin flip to have a 50/50 distribution of heads and tails. But what happens when you bet your money on it? Let’s see.

You decide to bet $1 on tails and see how things go. After 10 flips there have been 8 heads and 2 tails. You have lost $6 so far. Using your logic you would continue betting on tails because you expect it to even out to 50/50. After 100 flips there have been 55 heads and 45 tails. That’s a little closer to 50/50, but now you’ve lost $10 in the process. After 1,000 flips there have been 515 heads and 485 tails. Still closer, but now you’re down $30. After a grueling 10,000 flips there have been 5050 heads and 4950 tails. Now we’ve got a 50.5/49.5 ratio, which is darn close to 50/50, but you’ve lost $100. Even though the percentages are getting closer to 50/50, the actual number of flips is getting farther away and you’re losing money along the way.

Listen to your head and ignore your heart. I know that you want this to work, but I can tell that you know better.

-Sonny-

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You big spoil sport - now he'll start losing because you smashed his delicate belief system! zg

elegant attempt adventure boy! but no cigar :sad:
IMHO your thought process is first rate just applied against a hidden factor summarized by the capricious gambler's fallacy wherein our gift of recognizing patterns leads us astray when up against phenomenon shrouded by the confusion of dependent and independent events. the law of averages may be valid but alas anything can and will happen in the short term and with the odds stacked against us from the get go the short term can end up an eternity of hell
beware of smoke and mirrors
hopefully i can take my own advice lol cause this is a tough one for me as well.

Sonny said:

Your system is based on the Gambler’s Fallacy:

“If a fair coin is tossed repeatedly and tails comes up many times in a row, a gambler may believe, incorrectly, that heads is more likely on the following toss. This is an informal fallacy.”

http://en.wikipedia.org/wiki/Gambler's_fallacy

There is no reason to think that the wins and losses are going to immediately start evening out. In fact, it may take thousands or even millions of hands for the number to approach the expected ratio. What makes things worse is that even though the percentages are getting closer to what you would expect, the absolute differences will tend to get bigger!

For example, you would expect a fair coin flip to have a 50/50 distribution of heads and tails. But what happens when you bet your money on it? Let’s see.

You decide to bet $1 on tails and see how things go. After 10 flips there have been 8 heads and 2 tails. You have lost $6 so far. Using your logic you would continue betting on tails because you expect it to even out to 50/50. After 100 flips there have been 55 heads and 45 tails. That’s a little closer to 50/50, but now you’ve lost $10 in the process. After 1,000 flips there have been 515 heads and 485 tails. Still closer, but now you’re down $30. After a grueling 10,000 flips there have been 5050 heads and 4950 tails. Now we’ve got a 50.5/49.5 ratio, which is darn close to 50/50, but you’ve lost $100. Even though the percentages are getting closer to 50/50, the actual number of flips is getting farther away and you’re losing money along the way.

Listen to your head and ignore your heart. I know that you want this to work, but I can tell that you know better.

-Sonny-

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Well, I certainly would not predict when a winning hand is comming. I know I may lose 10-20 hands in a row at any given time. However, I do not believe that I can lose 100 hands in a row. As losses continue to rise and draw me away from the 48% win expectaion, I do believe it becomes more likley that the trend will reverse and start back toward expectation. The question is, how far can it go....and can my bankroll withstand it? To that end, I pose the following question:
In a game of 1000 hands, what is the most hands one could lose playing perfect basic strategy? We would "expect" to win 480. How far could we go below that? 380, 280, 180? I am not sure how to go about answering that question. It seems to me that if we go into a game knowing the minimum number of wins possible that you could curve fit a system to break even (or keep losses manageable) in the event that the worst happened.

Of course, I could be wrong....and probably am!

If you were to come up with a betting system that promised a profit if you win only any 28% of your hands in the long run,you would be on to something,provided you had the BR to back your play. Basing a system on winning 48% is a sure fire way to go broke.

adventureboy said:

I know I may lose 10-20 hands in a row at any given time. However, I do not believe that I can lose 100 hands in a row.

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It is rare, but it can happen. Just because something is unlikely doesn’t mean it is impossible. It’s safe to say that it probably won’t happen to you in your lifetime, but you can’t rule it out. It's going to happen to somebody eventually.

And besides, you don’t have to lose them all in a row. Most of the time your losing streaks will have a few wins and pushed mixed in. Your system is based on the total number of wins and losses so it doesn’t matter if they are consecutive or not.

adventureboy said:

As losses continue to rise and draw me away from the 48% win expectaion, I do believe it becomes more likley that the trend will reverse and start back toward expectation.

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There are three problems with that approach. First, you have no idea when the trend will reverse. The losing streak may continue for another 100 hands, or 1,000 or 10,000 before the results begin to shift back. As I said before, you have no reason to think that the reversal will happen now. The losses don’t have to be consecutive either, so the probability of a gradual losing streak is much higher than that of an uninterrupted losing streak.

But the trend doesn’t even matter. As I showed, the number of hands that you lose is likely to get larger even though the percentages are approaching the expected 48%. You are losing more money even though the results are slowly regressing. Even when your system works you are still losing money.

Lastly, the house still has the advantage over you. It doesn’t matter how many hands you lose, you are always more likely to lose the next one. Remember that the house gets its edge from the fact that you have to play your hand first. If you bust then you automatically lose even if the dealer busts too. That asymmetry is what makes the percentages unbeatable. It doesn’t matter if you lose 100 hands in a row, you still have to play your hand first so you are still a 48% underdog on every single hand that you play. If you include pushed then you are always a 43% underdog. In short, the trends can’t help you when the game is designed to favor your opponent on every single hand.

adventureboy said:

In a game of 1000 hands, what is the most hands one could lose playing perfect basic strategy?

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Well, you could lose all 1,000 couldn’t you? In fact, if you factor in splits and doubles then you could end up losing up to 8,000 bets in 1,000 hands. Obviously that would be very unlikely, but you still have to consider it.

adventureboy said:

We would "expect" to win 480. How far could we go below that? 380, 280, 180? I am not sure how to go about answering that question.

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What you are describing is the standard deviation from the mean. In this case the mean, or average result, would be 480. The standard deviation will tell you what sort of results to expect in the short term. It will give you an idea of how far above or below your results might be from the average. For example, it may tell you that for 1,000 hands you would expect a result of 480 wins plus or minus 36 hands. Here is a good introduction to standard deviation:

http://www.blackjackforumonline.com/content/Blackjack_Basic_Strategy_Betting_And_Risk.htm

That should explain how to find the numbers you are looking for. Feel free to ask any questions.

-Sonny-

Feeling free to ask a question...

Sonny said:

Here is a good introduction to standard deviation:

http://www.blackjackforumonline.com/content/Blackjack_Basic_Strategy_Betting_And_Risk.htm

That should explain how to find the numbers you are looking for. Feel free to ask any questions.

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Maybe I’m gonna have a breakthrough in understanding here!

I understand everything about the chart in that article, except where the SD% comes from. For example, that chart is based on 60 hands per hour. If I wanted to make a chart like that for 80 hph, where would I get the new SD% numbers from? If someone could explain that (as if to a small child, please), it would be wonderful.

Canceler said:

If I wanted to make a chart like that for 80 hph, where would I get the new SD% numbers from?

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Here’s a little more detail about what Snyder is doing in that article:

The standard deviation for one hand of BJ is about 1.15 units (I believe Snyder’s article uses 1.1 so we’ll stick with that for now). This is because of splits, doubles, blackjacks, and other things that skew the results. In order to find the SD for any number of hands, all you have to do is multiply 1.1 by the square root of the number of hands. In that article Snyder assumes 60 hands per hour so the SD for 1 hour of play is 1.1 * sqrt(60) = 8.52 units. Now we know that we will expect to be at –0.32 units plus or minus 8.52 units 68% of the time, and plus or minus 17.04 units 95% of the time.

If you want to use 80 hands per hour instead, just substitute 80 for 60 in the formula above. So for 1 hour of play your SD = 1.1 * sqrt(80) = 9.84 units.

Canceler said:

I understand everything about the chart in that article, except where the SD% comes from.

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The SD% is just the SD (in units) divided by the number of hands per hour (or action). At $80 per hour with a SD of 9.84 units your SD% = 9.84 / 80 = 12.3%.

-Sonny-

It works!

Thank you, Sonny! Now I'm finally getting it.

(The worst part is that in college I got a 4.0 in Statistics, but that was 30 years ago.)

Sonny,
Thanks for your responses...it is obvious your grasp of the game is far greater than mine. I will try one more question.

We all know the 48/52 is, in the long run, unavoidable. What I cannot quantify is the value of mean regression. For example, if I played 1000 hands and only won 200, I agree that the expectaion of the next hand being won is still 48%. However, I know this trend must reverse toward the mean. As you have stated, I just don't know when. Is there anyway to quantify the liklihood of the ievitable mean regression?

BTW--I have a friend who has a coin flip program. He ran a sim today for me with heads at 48% and tails at 52%. The # of flips was 1000, and he ran it 1000 times. The lowest heads won in the 1000 sessions was 223.

adventureboy said:

Sonny,
Thanks for your responses...it is obvious your grasp of the game is far greater than mine. I will try one more question.

We all know the 48/52 is, in the long run, unavoidable. What I cannot quantify is the value of mean regression. For example, if I played 1000 hands and only won 200, I agree that the expectaion of the next hand being won is still 48%. However, I know this trend must reverse toward the mean. As you have stated, I just don't know when. Is there anyway to quantify the liklihood of the ievitable mean regression?

BTW--I have a friend who has a coin flip program. He ran a sim today for me with heads at 48% and tails at 52%. The # of flips was 1000, and he ran it 1000 times. The lowest heads won in the 1000 sessions was 223.

Click to expand...

i think your question about when such a reversal or regression is likely to occur in a quantifiable way is interesting. i guess that is a rate of change sort of question which i guess would involve taking a derivative of what ever like in calculus i suppose. but if each hand is 48/52 then it doesn't seem you would have anything to differentiate. i suppose that in reality (which is unknown to us) that each hand is not really 48/52 but then we have no way of knowing when that is either lol. so even if each hand isn't really 48/52 the best that we can know is that each hand is apparently 48/52 lol since thats how it turns out in the long run. but i think differentiation requires a continuous function which apparently the phenomenon of wins or loss's in blackjack are not. so i guess the rate of change of wins to loss's and vice-a-versa are free to come in any order and quantity so that again it can't be differentiated or quantified.
not sure if i really know what i'm talking about here but it's something to ponder.

adventureboy said:

Is there anyway to quantify the liklihood of the ievitable mean regression?

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I could be way off here but I'll take a stab at your questions.

To the above, I'd say the liklihood is always 100%. Unfortunately there is no way to predict when that will occur so the knowledge is utterly useless.

To your question "In a game of 1000 hands, what is the most hands one could lose playing perfect basic strategy? We would "expect" to win 480. How far could we go below that? 380, 280, 180? I am not sure how to go about answering that question."

I'd guess about all one can do is quantify the liklihood of finishing with 380 wins or more (or less) in 1000 hands etc. So you could lose all 1000 hands but that has a very small probability.

Winning less than 380 hands is likewise, essentially, impossible, according to what I come up with. Which could be wrong lol. And then some lol.

But all you would really know anyway is the last 1000 hands were a billions to 1 event. In real llife, you'd probably also know the game is likely not fair.

So, if you want, it's always possible to determine the probability of winning at least x hands over the next however many hands should you feel that knowledge helps in any way.

This has nothing to do with units won/lost by the way - you might win 48% of the hands and they all might be 1 unit or 8 units lol.

In case Sonny cares, and even if he doesn't lol, since I'd like to know where I went wrong if I did, and I assume I did lol, I get a different stan deviation than he did of 15.8 (square root of 1000*.48*.52) for 1000 hands since all you seem to care about is simply whether a hand has a net win or loss and ties don't count, I assume.

What were the results of your 11,000 hands?

adventureboy said:

Is there anyway to quantify the liklihood of the inevitable mean regression?

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In theory, the regression is always happening. The problem is that it happens so slowly, and with such high variance, that it is difficult for us to see in the short-term. Here’s how most gamblers think of it: If you have a coin that lands on heads 10 times in a row, how do you know that it isn’t just regressing from 10 tails that happened before? How do you know that the coin isn’t at it’s mean after those 10 heads? Maybe it just “evened out” and now you’re betting on a completely random coin. You don’t know.

In reality, the fact that a coin lands on heads 10 times in a row doesn’t mean that the coin is “uneven” at all. It doesn’t mean anything. As Guynoire said, you would expect it to be 10 heads ahead for the rest of its life. If you flip that coin 100,000 more times it will average 50,010 heads and 50,000 tails. But, as the number of flips increases, those ten flip become less significant. The coin still may exhibit a bias, but the percentages become smaller as the number of flips increases. And don’t forget, the percentages can approach the expected 50/50 results even though the difference between the number of heads/tails is increasing.

This theory works in reverse as well. Imagine that the coin has been flipped 100,000 times before you flipped 10 heads in a row. You would expect 50,000 heads in the past, so now it has 50,010 heads and 50,000 tails. The coin is still at 50/50 and there is no reason to expect it to behave any differently than before.

-Sonny-

Kasi said:

…I get a different stan deviation than he did of 15.8 (square root of 1000*.48*.52) for 1000 hands since all you seem to care about is simply whether a hand has a net win or loss and ties don't count, I assume.

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But that doesn’t include splits, doubles and BJs. Instead of using .48*.52, try using 1.33 (the variance on a single hand of BJ).

-Sonny-

Sonny,

Well, I guess I will just spill the beans as to what I am really trying to get at. When counting cards, I experience a lot of the following:

1. +true count--lose hand after hand as the count goes up!
2. - true count--win hand after hand as the count goes down and I am betting the table minimum!

I know this happens to other counters. It is very frustrating to me...to sit there and count down deck after deck, finally get to a nice +6 true count in double deck, and wham!, the losing starts as the count continues to skyrocket! I was hoping to find some way to sidestep those freight trains as they barrelled down on me. Mean regression is what I came up with. Over the last few months I have still been counting, but only betting with the count when mean regression showed that it was neutral or in my favor. I would not increase my bet on a + count if my win ratio on hands was over 50%. This method has skyrocketed my win rate...but it looks like the consesus on this board (whick I really respect) is that I have just been lucky, so I should take the money and run. If regression analysis cannot stand on its own, I don't see how it can add anything to a counting method over the long haul. There is still a part of me that believes that there is something to mean regression, but I am not the smartest guy in the world so I am going to defer to the experts here.........unless so one out there would encourage me otherwise!!

adventureboy said:

Sonny,

Well, I guess I will just spill the beans as to what I am really trying to get at. When counting cards, I experience a lot of the following:

1. +true count--lose hand after hand as the count goes up!
2. - true count--win hand after hand as the count goes down and I am betting the table minimum!

I know this happens to other counters. It is very frustrating to me...to sit there and count down deck after deck, finally get to a nice +6 true count in double deck, and wham!, the losing starts as the count continues to skyrocket! I was hoping to find some way to sidestep those freight trains as they barrelled down on me. Mean regression is what I came up with. Over the last few months I have still been counting, but only betting with the count when mean regression showed that it was neutral or in my favor. I would not increase my bet on a + count if my win ratio on hands was over 50%. This method has skyrocketed my win rate...but it looks like the consesus on this board (whick I really respect) is that I have just been lucky, so I should take the money and run. If regression analysis cannot stand on its own, I don't see how it can add anything to a counting method over the long haul. There is still a part of me that believes that there is something to mean regression, but I am not the smartest guy in the world so I am going to defer to the experts here.........unless so one out there would encourage me otherwise!!

Click to expand...

If the count goes up during the actual hand you are playing you will probably lose. As an example you double your 10 or 11 and get the small card and the final card the dealer draws for their hand is low and they do not break. If however the count is going down as you play your hand then you are likely to win. When you double you get the big card on your 10 or 11 and the final card the dealer draws is high and breaks their hand. When we place big bets we are hoping big cards come out so we get our 20s and blackjacks, if they do not come out on the hand then we are probably in trouble.

If you are only raising your bets some of the time when appropriate for whatever reason you may turn an advantage to a disadvantage.

What you are betting on is for the count to return to normal distribution when all those 10s and As come out! LOL

adventureboy said:

Well, I certainly would not predict when a winning hand is comming. I know I may lose 10-20 hands in a row at any given time. However, I do not believe that I can lose 100 hands in a row. As losses continue to rise and draw me away from the 48% win expectaion, I do believe it becomes more likley that the trend will reverse and start back toward expectation. The question is, how far can it go....and can my bankroll withstand it?

Click to expand...

I was playing $25 a hand blackjack against the dreaded CSM's yesterday when anything can happen and went down from $1000 to $75 before things started to go right. I ended at $1300 1 1/2 hrs later... so a good result in the end considering bet size (up 12 units). I might have stayed longer had I not been due at work a half hour later.

So, answering your question I lost 37 units before things started to improve. Not all one after the other, but pretty damn close. I never got up in bankroll until the end of the session.

Standard deviation is just that - standard, worked out on millions on hands. In the short term anything is possible.

I will say however that my experience with hand-dealt or shoe games has never seen me lose that many in a row - I think the record number I've lost in a row (on a single hand up against the dealer) was 22. My maximum winning streak was 18 hands in a row. I only recall these because I diarised them at the time.

Sonny said:

That’s fine for even money bets, but that doesn’t include splits, doubles and BJs. Instead of using .48*.52, try using 1.33 (the variance on a single hand of BJ).-Sonny-

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OK - it just seemed to me all he cared about was achieving a net-win-of-any amount vs a net-loss-of-any-amount percentage and that how many units he may win on a win or lose on a loss was irrelevant to him as long as he achieved the winning percentage given his betting system.

So, essentially, an even money bet on a win vs a loss as opposed to how many units he may win or lose once he achieved those winning percentages. A biased coin with a 4% disadvantage if you will - after all he will "win" 48% and lose 52%.

Does this make any sense to you if all he cares about is he only "won" (a net win of any amount, say 1-8 units) 380 times in 1000 instead of 480? I just don't see it mattering to him, given the way he stated his betting system, that he might expect to win 1.2 units when he wins or lose 1.5 units when he loses kind of thing.

So I just don't see of how applying the variance of regular BJ in determining the liklihood of winning 380 hands out of 1000 vs an expected 480 of 1000 matters.

At least right now lol.

Kasi said:

So, essentially, an even money bet on a win vs a loss as opposed to how many units he may win or lose once he achieved those winning percentages. A biased coin with a 4% disadvantage if you will - after all he will "win" 48% and lose 52%.

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Ah, I understand. If his system is only counting winning hands vs. losing hands then the variance will be much lower. As you said, it would be like a biased coin. Actually it would be the same as the variance of red/black in Roulette, which is exactly 1. In that case the actual SD would be about 31.62 hands at the end of 1,000 trials. My numbers were for dollars, not number of hands.

However, I don't think those numbers are going to help him anyway. That kind of counting system is really not applicable to blackjack. Knowing the win/loss ratio doesn’t tell you anything about the results. You could win 4 hands and lose 6 hands, but if those 3 winnings hands were BJs then you would have broken even despite your 60% loss percentage. Conversely, you could win 6 hands and lose 4 hands but if your losses were splits or doubles then you would have lost 2 units even though you had a 60% win percentage. Knowing the distribution of wins and losses simply doesn’t tell you anything about the actual results of the game.

-Sonny-