This free course on blackjack and card counting was created by the GameMaster, publisher of the GameMaster Online website. It is reproduced here in its entirety with permission of the author. His 24-lesson course is an excellent introduction to winning blackjack.

To start at the beginning, visit the Welcome page.

### Expectation and Standard Deviation

If you flip a coin 100 times, your** expectation** is to receive 50 heads and 50 tails. But the reality may well be different; the measurement of that reality is called “standard deviation”.

**Standard deviation is a mathematical term used to predict the outcome of a situation.** In our coin-flipping exercise, we expect 50 heads and 50 tails to occur, but two-thirds of the time the actual result will be somewhere between 45 and 55 either way. That is, a result of 55 heads and 45 tails or something in between is not unusual; it will happen 68.3% of the time. That measurement is for 1 standard deviation from the expectation and if we were to run hundreds of ‘trials’ of 100 flips, we could plot our results on a bell curve and the vast majority of results would fall between 55 and 45 either way. What would be unusual would be to have a lot of trials where the result was actually 50-50! Got that concept in your mind? Good. You’ll need to understand this in order to survive the mental turmoil caused by the losses which are inevitable in this game.

**Nothing has caused counters to give up Blackjack more than a lack of understanding about normal, everyday standard deviation.** Counters who have trained hard unrealistically expect to win each time they play, so when they have several losing sessions, they forget what they’ve learned. Next thing you know, they’re over betting their bankroll and fail to play their hands properly and when they wake from the daze, their money is gone.

### PATIENCE AND SKILL WIN — HUNCHES AND WISHING WILL NOT WIN. PRAYER DOES NOT WORK AT BLACKJACK.

**So, what can you expect — what’s the worst which can happen? ** Well, you can lose all your money, but if you establish a bankroll of at least 50 ‘top’ bets, play proper basic strategy at all times and don’t over bet, you stand a good chance of making some $$$ at Blackjack — if the game at your local casino is a game which can be beaten. Did I ever say this was easy?

**The table below illustrates the possible results from varying hours of play at a fairly typical game.** Shown with the expectation are the possible dollar results as measured by 1 standard deviation (68.3% of the time) and 2 standard deviations which covers what will happen 95% of the time. Three standard deviations cover what will happen 99.7% of the time.

**Expected Win / Standard Deviation**

Assumptions: $12 average bet, 50 hands per hour,

1.25% average advantage.

Results | |||
---|---|---|---|

Time | Expected Win | 68.3% of the time | 95% of the time |

3 hours | $22.50 | +$191 to -$147 | +$360 to -$316 |

12 hours | $90.00 | +$428 to -$248 | +$766 to -$586 |

48 hours | $360.00 | +$1,036 to -$316 | +$1,710 to -$992 |

90 hours | $675.00 | +$1,600 to -$250 | +$2,525 to -$1,175 |

**Let’s talk about this a bit. If you were to play several hundred ‘sessions’ of 3 hours each, the average win for those sessions would be about $22.50.** (This comes from using a $5 to $60 betting spread which we discussed in previous lessons). But few sessions would result in a win of exactly $22.50; about two-thirds would be somewhere between a win of $191 and a loss of $147. Most of the other sessions could see you winning as much as $360 or losing as much as $316 and a few would see wins or losses even bigger than that!

Do you see now why it takes a bankroll of $3000 to support a $5 to $60 betting spread? In order to be successful, you must be able to absorb losses which are many times that of your ‘expectation’. These fluctuations are real; they will happen to you at one time or another and if you’re not prepared for them, you’ll either get frustrated and quit or lose your cool and blow your bankroll.

**Now look at the results for 90 hours of play. Most of you will be — at worst — about breakeven after that many hours.** A few might be up by $2500, but some of you could be down by $1175 or more. Boy, I’d hate to hear the names you’ll be calling the old GameMaster then! But it can happen and it won’t be unusual if it does, so ask yourself right now if you can deal with playing a disciplined game for 90 hours, still be at a loss and continue playing and betting as I’ve shown you. It’s sad, but most of you won’t be able to deal with that and you’ll be another victim of standard deviation. That’s why I’m not afraid of the casinos going out of business, even if every player in the world learns how to count cards — few have the patience to stick it out. I don’t want to be overly-negative, but that’s the reality. However, if you do stick it out, the percentages will eventually begin working in your favor. As I tell all my students, “the money comes in ‘chunks’ at Blackjack”. This is not a slow, consistent way to make money; your bankroll will, at times, resemble a roller coaster and it’s difficult to deal with that from an emotional point of view.

**Just try to understand the concept of standard deviation and continue ‘calibrating’ your eyes by doing deck estimation exercises with six decks.** As I’ve said before, you need to be accurate within a half-deck for computing the true count.

### Homework

Have a look around the blackjack knowledge base at this site. With over 200,000 posts about blackjack, that should keep you busy. ðŸ™‚

Please post your question on our message board forum for replies. Thanks.

Ken,

Expected Win / Standard Deviation. Assumptions: $12 average bet, 50 hands per hour, 1.25% average advantage, using be $5 to $60.

Under these conditions I could take a trip to play 90 hours with only 1,175 dollars. In the worst of the hypotheses would lose only 1,175, with good chances of winning back. Is that so?

Casino Rules:

8 decks, H17 , DAS, Late Surrender , Peek , 70 % penetration.

Ken,

I’m playing 8 decks, H17 , DAS, Late Surrender , Peek , 75 % penetration.

I threw 200 boxes, 20 boxes a day.

80 % of the time I played alone, just me and the dealer , the other 20 % at the maximum with another player .

My bets are fixed : $ 5 in true negative score or = 2. I use also the top ten and leave the table when the true count is too low.

The result so far is an average US 495.25 a day, I’m making 4925.50 in total.

I’m lucky?

I am within the standard deviation ?

I can improve my bets ?

Please do not consider.

Hi ken I’m playing in a casino where the rule are:

6 decks, h17, double any two cards, double after split, surrender allowed.

And: the casino have 2 jokers inside the 6 decks, and if in your hand, you have a joker, you win immediately, and the hand end.

I think with this I have pretty great advantage to my side, don’t you think.

Ken,

Expected Win / Standard Deviation. Assumptions: $12 average bet, 50 hands per hour, 1.25% average advantage, using be $5 to $60.

Under these conditions I could take a trip to play 90 hours with only 1,175 dollars. In the worst of the hypotheses would lose only 1,175, with good chances of winning back. Is that so?

Casino Rules:

8 decks, H17 , DAS, Late Surrender , Peek , 70 % penetration.

ken the general rule that your total br must be min 100 times your max bet and the trip br must be 20 times your max bet finds you ok?

and one crusial question !!!!! if my TOTAL br reduces at 50% by the bad variance should I resize my betting or continue like I still have all the money?

Ken,

I’m having trouble understanding the chart of standard deviation/ expected win. I am not sure if I’m getting this right. So after 3 hours of play, 68% of the time you will be either making $191 or losing $147, and 95% of the time making $360 and losing $316. I am not sure if this means that during the game, your net result can vary from 360 to -316, 95% of the time (ending with a profit of 22.5$) OR that after you finish a game you could either be at +360 or -316. So if you play 150 hands (3 hours with 50 hands each) and that there is a chance you could either be at +360 or -316, then there would not be a point in card counting since the advantage by counting cards is negligible as shown by this chart. Would you please help me understand what I’m missing or getting wrong ?

-Regards.

As the lesson describes, these numbers are the range of expected outcomes. 68% of the time, your result after 3 hours will lie somewhere between a loss of $147 and a win of $191. Let’s look at the 2-standard-deviation 95% range: somewhere between losing $316 and winning $360. In other words, if you played 20 three-hour sessions, then in 19 out of 20 of those sessions you would expect to fall somewhere in that range for that individual session. Even then, one time in the twenty sessions, you could expect to have more extreme luck, either good or bad. For that sessions, you would either lose MORE than $316, or win MORE than $360.

Of course, I must point out that even here, you can’t think in terms of exact outcomes. These are long-term averages. If you actually played 20 sessions, you may have no sessions that fall outside the 95% range, or you could have two or three sessions that do so.

As for the usefulness of counting, yes, the edge is small. In this case, the expected win is $22.50 for the three hour session. If you play a LOT of three-hour sessions with these criteria, your average win will be $22.50. But as the standard deviation numbers show, it won’t be a smooth ride. Sometimes you’ll win $300, sometimes you’ll lose $300. But notice that the ranges are shifted a bit, so that the highest expected wins are slightly bigger than the worst expected losses. Card counting is a long and bumpy ride, and the profits take time to overcome the luck factor.

Here i am again ken!!! ?

Great forum for once again.in the top of this page i see that in 90 hours the money goes from plus 2.500 to minus 1175.

Here is my guestion ….. is it possible the bad variance to win the maths and general to loose slowly slowly your bank roll entire?or the spread the correct count are suppurior than all the bad luck of the world.i cant imagine a player loosing 4000 dollars in a 3hours sessions slowly and in the end to fight with his last 1000.

Before some times you had told me that in the cards dont have memory who r u or how much u loose !!and u also told me that in the end things general meaning doesnt have to come even so a player take back his money.

Of course all these i am thinking maybe its just voodoo fear since cardcounting its a profitball art

ken please inform us when your new project will be ready !i wish all the best !

Ok ken i understand.thanks again for all your answers and hope to follow u in something new!!

Ok maybe i am the only idiot person in here but can u please give me in simple words the diffrence betwinn the meaning of the “standard deviation ” and the “variance”.and also the safety bankroll of the 3000dollars for spread of 5/60dollars make zero the percent of everything goes wrong and u loose all tour money or no?and if no what is the safety bankroll u need for never run out of money with 5/60spread

I’m tied up with a major move at the moment, and have no time to research each question that you arrive at, one by one. Get yourself some software and test these questions yourself. I’ve been pretty generous with my time so far, but I just cannot continue.

Note that the Knowledge Base here has a couple hundred thousand messages to peruse.

Hey Ken i am counting card for almost a year and i always prefer to play alone with the dealer. Lets assume that the true count is +4 and i would bet 40 pounds. Is it better to open 2 or 3 boxes instead of playing in one box? Will i reduce the standard deviation? And will i have more or less profit in the long run?

Wagering the same amount split up over two spots instead of one will decrease your variance. In that case, your expected win is still the same but the swings will be smaller.

Also, because the variance is lower, you can afford to make a larger total bet instead. If your bankroll allows you to make a $100 bet on one spot, you can afford to bet $75 on each of two spots and incur the same overall risk. With total action of $150 instead of $100, your win rate goes up but risk stays the same.

I’m in Brazil, where casinos are forbidden. So, I need to play in online casinos, using the live blackjack mode. The rules of the casino are: 8-deck game, cut off in the 50% of the shoe, stand on S17, double any 2 cards, double after split allowed, except for the double Aces (only one addition card). Blackjack pays 3:2.

My question is: With these conditions, is it possible to beat the dealer?

An 8-deck 50% game is very tough, but it could be beaten with a large enough spread if you are also willing to leave in any negative count. It’s going to be a frustrating grind, and because the edge will be so small, you could play perfectly for many hundreds of hours and still be behind due to bad luck. I wouldn’t waste time on this game.

Sorry Ken, one more question, so you can answer them at once.

You said in previous replies in this topic that someone should stand on a true count 4 and 15 vs dealer’s 10? That’s not what basic strategy says. I mean, basic strategy doesn’t correlate to counting in any way. So I am wondering, what do you actually mean? I didn’t read whole “GameMaster’s school ” yet, so i suppose these are some of the advanced strategies?

Best regards man, and i apologize for being boring.

Basic strategy says hit 15vT. But card counters who know the deck is full of high cards (with a true count of +4 or higher) can utilize that information to know that standing is now better than hitting in that case. Each strategy decision has a tipping point like this (an index number), where the strategy changes if the remaining deck composition is far enough from average. But by all means, focus first on getting basic strategy down cold. When I started, I used card counting for over a year with just basic strategy before I added strategy changes to my game.

Hello Ken.

My casino rules are : surrender(im not sure if it’s early or late), 6-deck game, stand on S17(not sure either), 3 splits allowed, double any 2 cards, double after split allowed. Blackjack pay 3:2.

AND: Blackjack with the same symbol of the cards (hearts for example) pays 2:1.

If the box has value of “21” with 7 cards in the box, it pays 5:1.

If there are three sevens with the same symbol in the box, it pays 5:1.

I think those are pretty good rules. Maybe even slight edge is on my side?

Can you tell me what should i change in my basic strategy because of the “AND” part above?

Also, my main question is:

Bet spread is 2$ – 100$ here. Can you tell me what is the minimum bankroll for a 3h play? I read in GameMaster’s school that it should be 50x top bet, but i suppose that’s because the bet spread there was 5$-50$ which is pretty bad. I don’t have 5000$ dollars. I have like 500$ maximum. I want to train for a couple of months, and then go to the casino, and see what can I do. I hope there won’t be that losing streak right away. But I’m fully aware of it.

And if I am able to stay disciplined, and focused, i will probably go in the casino with much bigger bankroll, after some decent time of practice.

But I’m asking you if 200-300-500$ is enough for this first time, with this 2$-100$ betting spread, or I should just wait, and save more money?

Sorry for the long post.

Thank you for everything.

Suited blackjack paying 2:1 adds 0.57% to the game. The base game you describe is around -0.58% if H17, or -0.36% if S17.

Suited 777 paying 5:1 adds only 0.008%, and 7-card 21 is probably less than that.

The overall net is a breakeven basic strategy game for H17, and +0.22% for S17 games. That’s a nice base advantage for the game.

The only strategy change is don’t split your 7s if they are suited. Hit for the possible 5:1.

If you are betting a top bet of $100, even if you are willing to take the risk of losing the bank, you need to have the table stakes to double and split as needed. Long term, I’m more conservative than the GameMaster. I advocate a full bankroll of 100X your top bet instead of his 50x. But for short-term plays, accepting the very high risk of losing your entire stake, I still would not sit down at the table with less than 10 times my expected top bet for the session. And I would never make a single bet that I couldn’t afford to split and double on the two hands.

First, do you have the numbers for the 99.7% of the time for 3 standard deviations, just out of curiosity?

What I’m really curious about for this lesson is how you would manage multiple sessions. Since in practice we’ll have to break for sleep, a 90 hour session isn’t realistic, so let’s use the 3 hours sessions as an example. If I save up $3000 and hit the casino per the rules in this example and after 3 hours I’m one of the unlucky ones that’s down $316, then I quit for the night and come back tomorrow. Do you recalculate your max bet and spreads based on a bankroll of $2684?

To go further, I could surmise that the 90 hour session doesn’t have to be continuous but could be thought of as 30, 3 hour sittings during which you mentally consider it that you’re on one long session “with breaks.” At 90 hours would you then recalculate your max bet and spreads for a bankroll of $1825? As you can see this could be able to expand to infinity (if you’re down $1175 after 90 hours, just consider it a low point on a 1000 hour session that you could end up ahead at the end so just keep paying in your initial calculations for a $3000 bankroll).

I know you should never limit your wins, so going up this isn’t a problem and it makes sense to figure in winnings from your last session to optimize each new one, but when you’re going negative there’s a finite bottom. If I’m saving $3000 to start out and I’m going down, is there a “walk away” point? Do I have to keep saving my paychecks to keep coming back up to 3000 until I finally get some sessions in the positive?

I ran the numbers of the 3 standard deviation for you, using the same criteria as the lesson.

The results:

3 hours: +$530 to -$485

12 hours: +$1104 to -$924

48 hours: +$2388 to -$1668

90 hours: +$3452 to -$2102

However, I recently discovered that the GameMaster’s numbers were derived in a different way than I had assumed, and unfortunately they now need a caveat… He calculated the variance here by just using the average bet size and the formula for SD. This causes his results to underestimate the volatility of this betting scheme, because there is more variance in a bet spread that averages $12 than in a straight flat bet of $12. This page really needs a thorough reworking as a result. It’s on my list.

As for how to treat an extended amount of play, if you are starting with a small bankroll you rarely have the ability to scale down your bet sizes if you are losing, because the game doesn’t stay profitable with smaller spreads and you are likely already at the limits of your ability to spread because of the small bank. For most beginning players who would be willing to raise another bank if they lose this one, resizing isn’t a realistic option. Instead there only real option is to keep playing as long as they can afford the bet spread and safely cover any doubles and splits that arise.

(Things look quite a bit different if you are dealing with larger bankroll numbers. Then you have some flexibility to resize as needed to reduce the risk of ruin.)

I played DD hit S17 or S17 any casinos but i won a little and down a lot of money.I dont know whats going on i played basic strategy counting cards Hi Low counts.But the problem the positive counts i cant get my big bets and my cards is always bad,

Iam thinking to stay away from the casino.Do i need not to stay my hard 14 or hard 15 against the dealer 20 if the count is +4TC

which remaning 2decks?or less?And how much do i need money to play $50 minimum bet in 1 session?thanks

The advantage in counting is quite small, so losing streaks are inevitable even if you are playing correctly. It is especially frustrating to have a losing streak at the very beginning, because you have nothing to compare it to. I would expect that most counters that start off with poor results give up the game. Players who begin with a winning streak instead will eventually have the same losing streaks, but they will have also experienced the other side of the coin which makes it far more tolerable.

You ask about standing with hard 14 or 15 vs a dealer 20. I assume you mean a dealer ten up. (Stop assuming that the dealer has a face card underneath. It is not true, and not helpful.) You confusingly mention a true count of +4 but then ask about remaining decks of 2 or less. You should stand with 15vT at a true count of +4 or more no matter how many decks are left. The conversion to a true count already takes into account the number of decks left. For example, with 2 decks left, that means you need a running count of +8 or more to stand with 15vT. As for the other hand, you should not be standing with 14vT. Just hit it.

Time away from the game could provide a much needed chance to improve your understanding before you go back, particularly if you are playing a $50 minimum bet game.

How much money is needed for a session at a $50 table? Well, that depends on how long you play and what your larger bets are. For an hour or so, you should have no less than $2000 or $3000 available. Preferably more, because you will often run out at that level. You need to be spreading to a top bet of at least $400 to make this game viable. I would prefer a session bankroll of $5000 for that game.

Ken ,

They are not 200 hands but 200 shoes.

My balance is too loud ?

This above standard deviation ?

OK, 200 shoes makes a lot more sense.

You’ve still been very lucky, but you are well within the range of possible outcomes.

As a guess, I would put your expected win somewhere around $1300.

You would need a simulation to get specific numbers.

Ken,

I’m playing 8 decks, H17 , DAS, Late Surrender , Peek , 75 % penetration.

I threw 200 boxes, 20 boxes a day.

80 % of the time I played alone, just me and the dealer , the other 20 % at the maximum with another player .

My bets are fixed : $ 5 in true negative score or = 2. I use also the top ten and leave the table when the true count is too low.

The result so far is an average US 495.25 a day, I’m making 4925.50 in total.

I’m lucky?

I am within the standard deviation ?

I can improve my bets ?

sorry , when the true count and more than 2 I bet $ 50 .

You have played a total of 200 hands and you’re up almost $5000, when betting $5 in bad counts and $50 at true +2 or higher?

That is almost impossible.

You ask if you have been lucky. Well, your expected win on 200 hands of this game is probably $10 or less. Does that answer your question? ðŸ™‚

By asking this question, it is clear that you have not read much about blackjack. Get yourself a decent book and understand it before you go back.