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.

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?

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.

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.

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.

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.)