Highly recommended. Foreword by Don Schlesinger. John Auston is the author of Blackjack Risk Manager software. To produce the tables for Chapter 10 of Don Schlesinger’s Blackjack Attack, Auston ran over 100 separate computer simulations of the hi-lo count, each 400 million hands, to test the effects of various numbers of decks (l, 2, and 6), with different rule sets. For more information Risk of Ruin click here. Round: In Blackjack, a Round starts with no cards on the table, and the player’s bets being placed. A hand is dealt to every player, and the dealer, and the Round ends when those hands have been played through, and the player’s bets have been paid out.

1. Blackjack Apprenticeship Basic Strategy
2. Blackjack Apprenticeship Chart
3. Blackjack Apprenticeship Risk Of Ruin Wisdom
4. Blackjack Apprenticeship Risk Of Ruin The Following

OK, I'll answer your question strictly in terms of math.

Betting style:
(martingale)
...
In addition...I would play 'never bust' - always force the dealer to make a hand AND beat mine.
...
First, what are the odds of losing 9 straight hands where you never bust.

The odds of winning with the 'never bust' strategy are approximately equal to the odds of being dealt either a 19-21 or 2-11 and upgrading to 19-21, plus the odds of dealer busting. You will win approximately 40% of hands and lose about 50%. Pushes not counting, you will lose about 55% of hands and win 45%.
The odds of losing 9 hands in a row are 0.55^9=1/217. The probability of a 9-hand losing streak can then be found as 1-(1-1/217)^(N-8)...
edit: Nevermind the formula. I've been told this calculation for the risk of a losing streak is oversimplified, and seems to double-count longer streaks. An accurate calculator can be found here: http://www.pulcinientertainment.com/info/Streak-Calculator-enter.html
The correct probabilities are 10.6% in 60 hands, 21% in 120, 39% in 240, 65% in 500 and 88% in 1,000 hands.
---
However, this should be put into context for comparing with other betting patterns. Here is a post I recently wrote elsewhere about martingale, I'll repost it here, tweaked a bit for context.
---
While most betting systems are mathematically neutral, martingale stands out as being mathematically damaging to the player in all long-term performance metrics, such as risk of ruin, SCORE, time to double the bankroll, and, critically, chance to double the bankroll.
For instance, the risk of ruin in typical blackjack with a 64-bet bankroll is 10% in 1,000 hands, 1.8% in 500 hands, or 0.01% in 200 hands. A 6-step martingaler will run out of his 64 bets the first 6-loss streak he gets. The probability of a 6-loss streak in fair coin flip is 1/64 (or 1/45 in blackjack), and a streak can begin on any hand.
So, it will take only 50 fair coin flips or 36 hands of blackjack to provide a 50% risk of ruin with 6-step martingale. A 10% risk of ruin is reached in a mere 10 hands. A 1.8% chance will be exceeded in just 6 hands, since your first 6-hand sequence entails a 2.2% risk of ruin. That is for a bankroll that will last flat-bettors through thousands of hands.
All this while, martingale limits the winnings to a single unit at a time, slowing down the winnings. Even under ideal conditions, perfect 1-0-1 (just what martingale is designed for), a 6-step martingaler needs 128 bets to double his bankroll, a 86% risk or ruin in coin flip or 94% in blackjack.
So while per-bet house edge is unchanged, with a martingale the chance to double a 64-bet bankroll is a mere 14% in fair coin flip, as opposed to 50% for a flat-bettor. This is a mathematical disadvantage, voluntarily creating house edge even in a game that doesn't have any. All martingale provides in the long run is just massively increased risk of ruin, without a corresponding increase in gain.
---I agree that a lot of people have seen the Martingale and variations before. However, not everyone has, and I think it's on the regulars to be able to explain gently why it's a non-working strategy, based on the questions the player asked.
Or keep schtum.
Your ire can be reserved for the point when the poster has revealed themselves to be willfilly ignorant/selling snake oil/unable to follow a logical train of thought.
Both questions could have been answered with some math, and it might just have been that the math would have been enough to convince the OP why it's a bad idea (TM).
As with all forum, what's old too one person is brand new to another, and repeated questions and themes will always appear. Or the forum disappears up its own backside into a insular community of anti-social jackarsery.

Well, you did offer some pretty ridiculous advice--'wait until the table gets hot.' Anyone offering such advice might very well have to be reminded that there is no such thing as a 'hot table', in the meaning of 'the players have recently won, so the players are more likely to win in the immediate future.'

Yeah - I didn't say 'wait until the table gets hot' or anything of the sort. I said that I've never seen a person make 12 passes in a row. I also said that while it is possible, in my 40 plus years of playing craps I've never seen it. Of course the dice don't remember but craps is a very simple, binary game. It is biased to the dark side. Even the house edge shows that. (And although people scoff at small biases I do not. Small errors accumulate into large errors, small advantages accumulate into large advantages. And even if that advantage is on the losing side I will lose less if I play the don't. That is just a cold, hard mathematical fact).
I also said that I have never seen more than 8 field numbers rolled in a row and while I am certain that it has happened I am also certain that it doesn't happen very often. I am also certain that for every set containing 8 field numbers rolled in a row there has been at least one set of 7.n non-field numbers rolled in a row (there being fewer non-field numbers than field numbers). I am also certain that craps is a closed system and that it contains a small number of events and that it regresses to the mean a lot more often than many people credit it with doing.
So if you are going to quote me try actually reading what I say and quoting me accurately. I think I have had the EV Knighthood up to my ass and beyond and I should be doing better things with my life. So if you will pardon me I will leave you now - for good.

Both questions could have been answered with some math, and it might just have been that the math would have been enough to convince the OP why it's a bad idea (TM).

But why should anyone bother to do the math? It's like resorting to a detailed explication of physics and chemistry to show someone why their scheme to turn cotton balls into plutonium won't work.
It's a far better service to simply say to such a person, 'It won't work.' If you explain the math, and by some miracle that person understands that math and agrees with the conclusion, they'll just go back to their basement and cook up some different system in the forlorn hope that the math will validate that new one.
I think the odds of the math convincing the OP that Martingales don't work were about 40,000,000 to one. I respect the various quixotic tries to do so, though.

OK, I'll answer your question strictly in terms of math.
The odds of winning with the 'never bust' strategy are approximately equal to the odds of being dealt either a 19-21 or 2-11 and upgrading to 19-21, plus the odds of dealer busting. You will win approximately 40% of hands and lose about 50%. Pushes not counting, you will lose about 55% of hands and win 45%.
The odds of losing 9 hands in a row are 0.55^9=1/217. The probability of a 9-hand losing streak is 21% in 60 hands, 40% in 120 hands, 54% in 180 hands, 65% in 240 hands, 90% in 500 hands, 99% in 1,000 hands. The formula is 1-(1-1/217)^(N-8), where N is the number of hands played.
---
However, this should be put into context for comparing with other betting patterns. Here is a post I recently wrote elsewhere about martingale, I'll repost it here, tweaked a bit for context.
---
While most betting systems are mathematically neutral, martingale stands out as being mathematically damaging to the player in all long-term performance metrics, such as risk of ruin, SCORE, time to double the bankroll, and, critically, chance to double the bankroll.
For instance, the risk of ruin in typical blackjack with a 64-bet bankroll is 10% in 1,000 hands, 1.8% in 500 hands, or 0.01% in 200 hands. A 6-step martingaler will run out of his 64 bets the first 6-loss streak he gets. The probability of a 6-loss streak in fair coin flip is 1/64 (or 1/45 in blackjack), and a streak can begin on any hand.
So, it will take only 50 fair coin flips or 36 hands of blackjack to provide a 50% risk of ruin with 6-step martingale. A 10% risk of ruin is reached in a mere 10 hands. A 1.8% chance will be exceeded in just 6 hands, since your first 6-hand sequence entails a 2.2% risk of ruin. That is for a bankroll that will last flat-bettors through thousands of hands.
All this while, martingale limits the winnings to a single unit at a time, slowing down the winnings. Even under ideal conditions, perfect 1-0-1 (just what martingale is designed for), a 6-step martingaler needs 128 bets to double his bankroll, a 86% risk or ruin in coin flip or 94% in blackjack.
So while per-bet house edge is unchanged, with a martingale the chance to double a 64-bet bankroll is a mere 14% in fair coin flip, as opposed to 50% for a flat-bettor. This is a mathematical disadvantage, voluntarily creating house edge even in a game that doesn't have any. All martingale provides in the long run is just massively increased risk of ruin, without a corresponding increase in gain.
---

nice post

Of course - how ignorant of me to forget that single, most important aspect. Oh thank you wise one for setting me on the path to enlightenment.

What did I do?

But why should anyone bother to do the math?
It's a far better service to simply say to such a person, 'It won't work.' If you explain the math, and by some miracle that person understands that math and agrees with the conclusion, they'll just go back to their basement and cook up some different system in the forlorn hope that the math will validate that new one.

I disagree with your comments for two reasons. I think that the reason this website exists is to educate and inform people. By just telling someone something won't work in answer to their question...

So what I can't get my mind around basically is...
First, what are the odds of losing 9 straight hands where you never bust.
Second, since extra profit will be made whenever I get a blackjack (and obviously, the farther into the sequence I am, the higher the profit), how significant is that to the overall final edge?
Any input would be greatly appreciated! I tested this method out on a free game online for around 3 hours (I know, small sample size for sure) and profited $435.

... you are in effect telling them that their question is not valid and is not worth answering. I assume that you have decided that the question is not worth answering mathematically mkl but please don't presume that others on this site feel the same way. I know you like to respond to every post on the site (or at least the overwhelming evidence points to that conclusion) but perhaps you might look at a post such as this one and simply decide not to post anything instead of jumping on it and insulting the poster.
I know (as does anyone who has read your posts) that you don't believe any kind of Martingale system can possibly create an advantage for a player. I agree with you as do most here. If you feel you've explained this to death and have no inclination to take the time to explain it again, you could just ignore the question.
My second point is that having read your posts in the past it seems to me that you are not sufficiently capable of actually performing the math to answer many of these math oriented questions. It's not that you can't add and subtract and multiply and divide; I'm sure you can. It just seems that the breaking down of the questions to be able to create a workable formula is a bit over your head from time to time. Rather than leave the question for someone better suited to provide an answer, you prefer to give some half-hearted quasi-mathematical answer and then deride the person who has asked the question.
I'd say that's what's happened here.
To the OP (Bruski), I'm working on an answer.

Blackjack probability is just like any other probability in the casino. It’s a means of measuring the likelihood of certain events. You’ll often see probabilities expressed as percentages, but they can be expressed as fractions or odds, too.

Blackjack statistics is a way of measuring your actual results and comparing them to your predicted results. In the long run, your actual results will start to resemble your predicted results. But in the short run, random chance will ensure that anything can happen.

That’s why some players have huge losing streaks, while others have big winning streaks. The casino doesn’t worry about this, because they’ve set up the games and the payouts in such a way that they’re ensured a profit in the long run. That’s a matter of expected value.

Some Definitions Related to Probability and Expected Value

In fact, that’s probably the best way to introduce this blog post—with some definitions of some terms related to blackjack probability in general. That way you’ll be able to dig deeper into the main points of the post below.

Let’s start with the phrase “probability.” The word has 2 meanings. The first is that probability is the branch of mathematics that deals with the likelihood of an event occurring. The 2^nd is more useful—probability also refers to an event’s likelihood.

Probability is measured numerically, and an event’s probability is always a number between 0 and 1. An event with a probability of 0 will never happen. An event with a probability of 1 will always happen.

An event with a probability of 50% will happen half the time, on average. 50% is one of the more common ways to express that probability, but you could also say that this event has a probability of 1/2 and still be correct. Another useful way to express probability is in odds format. That’s when you compare the number of ways something can’t happen with the number of ways it can happen. With a 50% probability, an event has “even odds,” or 1 to 1 odds.

Expressing probabilities as odds can be useful when trying to decide whether you have an edge or not. In most casinos, the games all have a built-in edge, but blackjack is exceptional in this respect. I’ll get into that a little later in this post.

Another important concept in gambling probability to understand is the concept of “expected value.” This is what a bet is “worth.” A bet’s expected value can be positive or negative, but if you’re a player in a casino, it’s almost always negative. The formula for expected value is simple, too:

You multiply the probability of winning by the amount you stand to win. You also multiply the probability of losing by the amount you stand to lose. You subtract one from the other, and you have the expected value of the bet.

For example, if you have a 50% chance of winning $1, and you also have a 50% chance of losing $1, you have an expected value of 0. That bet is a break-even bet; over time, you won’t win any money at it or lose any money at it.

But let’s say you have a 45% chance of winning $1, and you have a 55% chance of losing $1. Now your expected value looks quite different:

+$0.45 – $0.55 = -$0.10

This means that over time you’ll lose 10 cents every time you make this bet.

Almost all casino game bets have a negative expected value. You’ll either lose more often than you’ll win, or you’ll win too little when you win to break even, or some combination of these factors. That’s how casinos stay in business.

That’s also why gamblers walk away a winner. In the scenario outlined above, you can’t lose 10 cents on a single bet or even a series of 2 or 3 bets. You’re going to win or lose $1 on each hand.

The expected value is an average expected over the long run.

And the long run is longer than most people think.

That’s why the casino can afford to pay winners occasionally and still make a huge net profit overall.

“The house edge” is another way of looking at the expected value of a bet, but it’s only used to describe bets where the casino has an edge over the player.

How the Casino Wins Consistently at Blackjack

You would think that the casino would have no edge in a game like blackjack. After all, the dealer is getting the same cards as the players. He has the same probability of being dealt a blackjack or going bust as a player.

The amazing thing about the house edge in casino games is that it’s usually a simple byproduct of the rules used by the casino for the game. For example, in roulette, the house gets an edge by paying off all the bets as if the 0 and the 00 weren’t on the wheel.

In blackjack, the house gets its edge by making the players resolve their actions and bets first before the dealer acts. In other words, you must make all your playing decisions before the dealer ever acts. This means that if you bust (get a total of 22 or higher), you automatically lose your bet—even if the dealer also goes bust. Since you acted first, and the dealer resolved your bust before having to play, the house has an advantage.’

This is a huge advantage made bigger by the fact that some players don’t play their hands optimally from a mathematical standpoint. In many cases, the best play is to stand on a hand which isn’t likely to win unless the dealer busts. A lot of players have trouble with this.

This advantage is so big for the casino that it can even afford to offer an extra high payout on some hands. In most casinos, a 2-card hand totaling 21 (a “blackjack” or “natural”) pays off at 3 to 2 odds. This means if you bet $100 and get a blackjack, you win $150.

The casinos can afford this bonus payout and still have a profitable mathematical edge over the player. This 3 to 2 payout is one of the reasons that smart players can get an edge over the casino, and I’ll have more to say about that later in this post.

Since there a finite number of cards in a blackjack deck, it’s possible to calculate the mathematically best play in every situation. This is called “basic strategy.” Computer programs analyze the potential results of every possible decision in every possible situation. The move with the highest expected value is the correct playing decision.

The average blackjack player loses an average of 5% of every bet he places at the blackjack table. The average blackjack player is playing with “common sense,” “hunches,” or just pure dumb instinct.

The smart blackjack player, though, memorizes and uses basic strategy in every situation. This reduces the house edge to less than 1%. Depending on the rules variations in effect at a specific blackjack table, the house edge might be significantly less than 0.5%.

But it doesn’t matter how low the house edge is. If the house has an edge over the player, if the player gambles long enough, he’ll eventually lose all his money. That’s how the casinos stay in business.

Blackjack, though, is different from almost every other game in the casino. It’s a game where a smart player with the right strategy can get an edge over the casino. This is beyond the abilities of most players, and even a lot of players who THINK they’re playing with an edge over the casino are mistaken.

I talk about why and how that is in the next section.

How Probability in Blackjack Differs from Probability in Other Casino Games

The reason a strategic player can get an edge in blackjack is because as each card gets dealt, the composition of the deck as a whole changes. In any random shuffle of a 52-card deck, the cards might fall in any given pattern. But sometimes higher-value cards and lower-value cards are dispersed in the deck unevenly.

When I say “higher-value cards,” I means 10s and aces. Since these are the only cards that can create a blackjack—and the corresponding 3 to 2 payout—it’s better for the player if there are a relatively large number of these cards left in the deck.

“Lower-value cards,” on the other hand, increase your probability of going bust when you take a hit. They also make it harder to hit your 3 to 2 payout on the blackjack. If a deck has a relatively higher percentage of lower-value cards in it, the casino has a bigger edge than usual.

This might seem obvious, but think about it this way if it still isn’t clear:

You’re playing blackjack, and over the course of the 1^st couple of hands, all 4 of the aces are dealt.

What is the probability of being dealt a natural after this?

Since you need a 10 AND an ace to get a natural, your probability of getting a natural drops to 0.

Here’s another way to think about it:

When you’re playing roulette or craps, the odds are the same on every outcome. That’s because the number of possible outcomes on a roulette wheel don’t change. You always have 38 numbers with an equal probability of coming up.

When you’re playing craps, those 2 dice have the same number of sides (6) every time you roll them.

You don’t eliminate a number from the roulette wheel once a ball has landed in that slot. You start over on the next spin.

Blackjack Apprenticeship Basic Strategy

You don’t eliminate a number from the sides of the dice just because it came up on the previous roll.

But when a card gets dealt in blackjack, it’s gone from the deck until the deck gets re-shuffled.

That changes the probabilities on every hand.

How You Can Use This Information to Get an Edge over the Casino in Blackjack

If you could bet more when the deck has a higher ratio of 10s and aces and bet less when it doesn’t, you could get an edge over the casino. You’d be putting more money into action when you’re more likely to get a 3 to 2 payout.

And as it turns out, you CAN do exactly that.

You’ve probably heard of “card counting.”

Unless you’ve read about it before, you probably think it’s beyond the capabilities of most mere mortals. Maybe you saw Rain Man as a teenager and think you need to be able to memorize every card as it’s played to succeed in counting cards.

But the truth is, anyone who can add and subtract 1 can count cards. Maintaining the level of concentration of keeping that count accurate while not looking like you’re counting is the real trick.

You don’t track specific cards when you’re counting cards in blackjack. You just track the ratio of high cards to low cards. You assign a value of -1 to the 10s and aces, for example. Then you assign a value of +1 to the 2s, 3s, 4s, 5s, and 6s. The 7, 8, and 9 have a 0 value each.

When the running count is positive you bet more. The higher the count, the more you bet.

When the running count is 0, you bet less.

By doing this, you get a mathematical edge over the casino.

You can also use the count to inform your basic strategy decisions. Basic strategy assumes a full 52-card deck, but once the ratios of cards change as the deck gets depleted, the playing decision with the highest expected value can change.

This can increase your edge over the house even further.

You don’t gain a huge edge over the casino through card counting, though. You usually wind up with an edge over the casino that’s close to the edge the casino has over the basic strategy players.

This means that you might be playing with an edge over the casino of 0.5% or 1%.

That’s not a huge edge.

But it’s big enough.

Projecting Hourly Win Rates, Loss Rates, and Bankroll Requirements

How much does the average blackjack player lose per hour?

Blackjack Apprenticeship Chart

The formula is simple:

It’s the number of bets per hour, multiplied by the average size of those bets, multiplied by the house edge.

At an average blackjack player, you might see 80 hands per hour. Say you’re betting an average of $5 per hand, and you’re not using basic strategy, you’re looking at $400 in action per hour and losing 5% of that–$20/hour.

A basic strategy player, on the other hand, might reduce the house edge of 0.5%. This reduces his hourly loss rate to just $2/hour.

A card counter might be operating with a 1% edge over the house though. At the stakes we’re talking about, he’s winning $4/hour.

But consider this:

He’s raising the size of his bets based on the count, so his average bet size won’t be $5. It’ll probably be closer to $20.

Now we’re looking at $16/hour.

That’s not a great living, by the way. You can probably make the same money working at an In N Out Burger, in fact.

But as your bankroll grows, so does the average size of your bets. The people making real money counting cards might be putting $100 per hand into action on average. That’s $8000 per hour, or $80/hour in winnings.

$80/hour is some real money.

But one thing card counters need to think about is something called “risk of ruin.”

Remember how I talked about short term variance, and how you can’t expect long term expectations to hold true in the short term?

Just because you have an edge when counting cards doesn’t mean you’re going to show winnings every hour. You’re going to have wild swings of luck. That 1% edge is an average over thousands of hands.

You need a big enough bankroll to handle those swings in fortune without going broke.

The bigger your bankroll, the less likely you are to go broke before your edge and the long term kicks in.

This probability is called “risk of ruin.”

The most conservative approach is to have 1000 units to bet with. If you’re averaging $100 bets, you need a bankroll of $100,000 to play. If you have that kind of bankroll, your risk of ruin is just 1%.

On the other hand, if your tolerance for risk is better than that, you could get away with a much smaller bankroll—maybe 200 units. You’d still need $20,000, but you’d be able to play at that level. Your risk of ruin goes way up, though—to 40%.

I suggest to Texas holdem players that they know they’re good enough to move up in level when they can increase their bankroll to the appropriate amount for that new level. I think this recommendation holds true for blackjack players, too.

Blackjack Apprenticeship Risk Of Ruin Wisdom

If you have $2000, you should be playing for an average of $10/hand. If you’re succeeding at that level, you’ll eventually have a bankroll of $4000, and you can move up to $20/hand, and so on.

How conservative or aggressive you are is up to you and your temperament.

Conclusion

Blackjack probability is a fascinating subject with no end of subtopics you can discuss. I could just as easily have written about the probability of going bust with certain tables as I did with the approach I took. I just thought it would be more useful to tackle the subject of blackjack statistics from an aerial view.

Most people, frankly, aren’t cut out to be card counters. It sounds easier in theory than it is in practice. Blackjack in most casinos is fast-paced and confusing, especially if you’re new to the game. It’s hard to keep up with those numbers in your head without looking like you’re paying too much attention.

And don’t forget that part:

The casino is watching for card counters. To say they frown on counting cards is an understatement. Casinos will risk throwing out players they could profit from if they suspect them of counting cards.

In fact, I think most casinos would be better off if they lightened up on card counters. I know plenty of would-be card counters who make enough consistent mistakes that they only THINK they’re playing with an edge over the casino.

Blackjack Apprenticeship Risk Of Ruin The Following

My guess is that the number of would-be card counters who are profitable in the long run make up between 5% and 10% of the total number of card counters in the business.

At any rate, knowing something about the probabilities behind the game makes it more fun, even if you have no interest in being an advantage player.

And if you’re not an advantage player, fun’s what it’s all about. Comparing the cost of that fun to the amount of enjoyment you get from playing is what smart recreational gambling is all about.

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