Standard deviation is the square root of the number of trials, divided by 2. It's not difficult to figure out on any pocket calculator that has a square root key. The square root of 100 is 10, so the standard deviation on 100 trials is 10/2 = 5. The square root of 1,000 is approximately 31.6, so the standard deviation on 1,000 trials is 31.6/2, or approximately 15.8.

Once you understand what the square root of a number means to a statistician, you will understand why it is perfectly normal for you to come up with 7 heads on 10 flips of a coin, but nearly impossible for you to come up with 7,000 heads out of 10,000 flips of the same coin.

The standard deviation of 10 = 3.16/2 = 1.58.
The standard deviation of 10,000 = 100/2 = 50.
So, to come up with heads seven times in ten flips, is to be 2 away from the expectation of 5, and well within two standard deviations (3.16).

But to come up with heads on 7,000 of 10,000 flips, is to be 2,000 heads over the expectation of 5,000. And since one standard deviation on 10,000 flips is only 50, this result is 40 standard deviations away from what's expected. Statistically, this is nearly impossible.

Just how impossible is it? Statistically, we expect to be within one standard deviation 68% of the time, within two standard deviations 95% of the time, and within three standard deviations 99.7% of the time. Suffice it to say that if we get a result that is 40 standard deviations away from our expectation, either the coin or the flipper is crooked. You have a much better chance of winning your state lottery than you do of flipping 7,000 heads in 10,000 tries with an honest coin.

All blackjack players must be concerned with normal fluctuations, as they are a crucial factor in the size of the bets you can afford to make. The following guidelines are based upon statistical realities that should be more than enough for most casual players.

In an hour of play, or about one hundred hands, in a dead even game, you generally will not be ahead or behind by more than 20 units. On rare occasions, however, in a single hour of play, you may expect to be ahead or behind as many as 35-40 units.

If you play off and on over a period of a few days—say, ten hours of play, or about a thousand hands—you probably won't be ahead or behind by more than 75 units, but on rare occasions, you might be ahead or behind by 120 units in a one-thousand-hand period. These estimates of fluctuation assume you always bet only one unit on each hand, and that neither you nor the house has any significant long-term advantage.

• Pseudo-blackjack Games (0)
• Blackjack – So How Much Money Do You Need? Part3 (0)
• Blackjack – So How Much Money Do You Need? Part2 (0)

So, let's look at some of the practical considerations of bankrolling your play. Essentially, what you are up against is what mathematicians call normal fluctuation.
As shown by our computer simulation results, even when you have a strong advantage over the house, you'll still sometimes lose, because in the short run, anything can happen.

This is true even for the casinos— although the house enjoys a large edge on their slot machines, on any given day some slot players will win more than they lose, which is why people return to the slots. If all slot players lost every time they played, no one would play. All casino games are designed to allow players to go home winners fairly regularly—just not often enough to compensate for their long run losses.

Let's stick with blackjack, though. Assume you learn to play basic strategy, so that you nearly eliminate the house edge. How much can you win or lose due to normal fluctuation?
To answer that, start by imagining that all of your bets are of equal size. Rather than assigning some dollar value, let's say instead that you bet one unit on each hand. We will assume you are in a traditional single-deck Las Vegas Strip game, playing perfect basic strategy, so that for all intents and purposes, the game is dead even. Over the long run, you'd expect to win nothing and lose nothing. It's like flipping a coin.

Of course, if you try flipping a coin a thousand times, and recording the results, you'd be highly unlikely to come up with exactly 500 wins and 500 losses. There are precise mathematical formulas for predicting the limits of normal fluctuation, and with an introductory course in probability and statistics, you would know how to make such estimations. But for now, let's develop some practical guidelines describing the best and worst you might expect due to normal fluctuation.

Statisticians use the term standard deviation to explain variations from the expected result. For instance, if you flip an honest coin 10 times, your expected result is five heads and five tails. If, however, you came up with 7 heads and 3 tails, this would not be indicative that the coin was dishonest. It would be considered a normal fluctuation.

However, if you flipped a coin ten thousand times, and it came up 7,000 heads and only 3,000 tails, it would be very unlikely that this was an honest coin. Even though the ratio of heads to tails has remained 7 to 3, the large number of tosses makes the result highly unlikely.

• Pseudo-blackjack Games (0)
• Blackjack – So How Much Money Do You Need? Part3 (0)
• Blackjack – So How Much Money Do You Need? Part2 (0)

You may note that the SPI Chart includes no entries for the new rule variations found in some of today's games. In the past few years, many casinos in Las Vegas have begun offering single-deck blackjack where naturals pay 6 to 5 instead of the traditional 3 to 2. I'll call these games "BJ Pays 6 to 5." This rule is so bad that there is little use devising an SPI value for it. The negative value would simply negate the possibility of any such game providing a card counter with a profit opportunity. Super Fun 21 is even worse than BJ Pays 6 to 5, and the host of rules favoring players are not strong enough to counter the negative effect.

Technically, both BJ Pays 6 to 5 and Super Fun 21 can be beaten with card counting strategies. The problem is that you need deeper penetration than most casinos provide, and you would also have to use a much bigger betting spread than you would need for a traditional BJ Pays 3 to 2 game. So, my advice is to avoid these pseudo-blackjack games altogether.

Likewise, some casinos now use continuous shuffle machines, or CSMs, on some tables. These machines effectively cut penetration to 0%, making card counting a complete waste of time since there's nothing to count! Stay away from tables that use CSMs.

Another disturbing development in Nevada in the past few years has been the introduction of "fake" one- and two-deck games. At these tables, a machine is used to shuffle six to eight decks, and then the dealer removes only one or two decks from the machine, dealing what appears to be a hand-held one or two-deck game. In fact, what you have is a six- or eight-deck game with extremely poor penetration. Counting cards in a game like this is a waste of time. So watch out for the fake blackjack that casinos keep introducing. All of them are sucker games.

• Blackjack – So How Much Money Do You Need? Part3 (0)
• Blackjack – So How Much Money Do You Need? Part2 (0)
• Blackjack – So How Much Money Do You Need? (0)

A few final comments on playing on a replenishable bankroll: If you want to take one month's win—after a successful trip—and add it to the amount you can afford to lose the following month, this will give you a lot more flexibility, to say nothing of total playing hours. In fact, I recommend that you do this if you are serious about making money from the casinos.

But never, never, borrow against next month's replenishable funds—that's a very bad habit to get into. In fact, about the only time I would recommend it would be when you need the extra funds to double down or split on a current hand in play. You always want to maximize your win potential in these types of situations, but if you lose those "borrowed" funds, then you must seriously consider whether or not you can play next month on your full replenishable amount. If the amount you go over your trip bank is negligible, forget about it. But if you lose half of next month's playing bank due to some incredible double/split opportunity that doesn't go your way, then seriously consider playing the following month on half of your normal allotment, or even waiting until you can add that allotment to the following month's playing bankroll, so you can play with more flexibility then.

Regardless of how big or small your annual replenishable bankroll is, you should be able to draw up a bet-sizing schedule using the chart above as a guideline. Do not waste your time studying complicated Kelly betting strategies or risk of ruin charts, or any of the complex methods that pros on fixed bankrolls use, or teams of pro players seeking investors must master. With a replenishable bankroll, you're well on your way to making money at this game. But if you quit your job, move to Vegas, and start playing full time on the fixed amount of capital you arrived with, you've got some serious considerations ahead of you.

• Pseudo-blackjack Games (0)
• Blackjack – So How Much Money Do You Need? Part2 (0)
• Blackjack – So How Much Money Do You Need? (0)

If you're working a job that pays $40K per year after taxes, and you figure your real living expenses to be about $30K per year, then you've got a replenishable bankroll of about $10K per year. It may only be $833 per month, but in fact, you can play as if you've got $10,000 behind you, provided you strictly adhere to never losing more than your monthly replenishable amount. This means that in many months you will be put out of commission at the tables very quickly due to negative but entirely normal fluctuations. But you'll compensate for that when you win inordinate amounts, wins that never would have been possible on an $833 bankroll.

Just about every book on card counting addresses the bet-sizing question from the perspective of the player who is on a fixed bankroll, even though most players, including many semi-pros and serious high-stakes players, operate from replenishable funds. If you have an annual replenishable bankroll of $ 10,000, you will never make it as a card counter if you always play as if you have a total bankroll of $833, since that is the actual amount in your pocket. If you're going to make it as a counter at a serious level of play, then you will have to have the guts to bang out some bets that could easily break you on that trip if the flux go against you, and you will have to have the discipline to quit, on that trip, when that occurs.

My advice for a player on a replenishable bankroll is very simple. Always play as if your total bankroll is the annual replenishable amount, and play at the one-quarter-Kelly level. This boils down to a fairly simple chart: The percentage of your replenishable bank that you bet is always one-quarter of your % advantage (or true edge). Whether you use the Red Seven Count, the Hi-Lo, or the Zen Count, it looks like this:

Annual Replenishable Bankroll
True Edge $2.000 $5.000 $10.000 $20.000
1% $5 $12 $25 $50
2% $10 $25 $50 $100
3% $15 $35 $75 $150
4% $20 $50 $100 $200

This means that if you've got that $10K replenishable bankroll, you can actually bet from $25 to $100 on your high counts, even though you may not be going into the actual game with much more than $800 bucks in your pocket. When you've only got $800+, bets of $100 are close to suicidal. What if you have to double down?

What if you have to split, and re-split, and then double after you split? You can lose half or more of your trip bankroll on a single round of play! Mitigating this risk is the fact that 4% advantages very rarely occur. You won't often place $100 bets in today's blackjack games. But when these strong advantages do occur, then bite the bullet and bang it out. This is no time to be chicken; these are the types of opportunities that you're looking for as a card counter. If you lose your trip bank, so it goes. If you've got enough hours at the tables, maybe the casino will comp you to dinner. There's always next month, and in the long run, you'll get them more than they'll get you on these advantage plays. You either have the heart for this game or you don't.

• Pseudo-blackjack Games (0)
• Blackjack – So How Much Money Do You Need? Part3 (0)
• Blackjack – So How Much Money Do You Need? (0)

We still haven't answered the question that is the title of this chapter. Most players would consider it a waste of time to make $10 bets when they arrive at a casino with $2,500 to play with. In fact, very few players would do this—if you're going to gamble with that amount of money, you generally want to see a meaningful result. My ultra-conservative betting advice above, though, assumes that you have a fixed bankroll of funds that are not easily replenishable. For most players, that's not the case.

A replenishable bankroll has no fixed dollar amount, and most players, in fact, play on replenishable bankrolls. There are various common types of replenishable bankrolls, the most common being a job. If you are currently earning an income from some non-gambling source, and some portion of this income is expendable, in the sense that it is money you would normally spend on recreational pursuits, hobbies, entertainment—your disposable income money—then so long as you stay employed, that's your replenishable bank. If you deliver pizza for a living, then this bankroll may be less than $100 per month, if you're a dentist, it may be quite a few thousand dollars more. Much depends on your lifestyle and obligations—if you are a married dentist with five kids and you're paying off a mortgage, the pizza delivery boy may actually have a larger replenishable bankroll than you do.

I will leave it to you to figure out exactly how much of a replenishable bankroll you have, but here is the important distinction between a fixed bankroll and a replenishable one: A replenishable bankroll is a specified number of dollars that replenishes itself over a given time period. For instance: $300 per month, or $1,000 per week, or $52,000 per year. It does not include your life savings, or some amount of money you can borrow against your credit cards.

Compared to a fixed bankroll, a replenishable bankroll is an absolute joy for a card counter. With the replenishable variety, you can never go broke from negative flux. You may be put out of commission for the rest of the week, but come next month and a couple more paychecks, you're back in business.

• Pseudo-blackjack Games (0)
• Blackjack – So How Much Money Do You Need? Part3 (0)
• Blackjack – So How Much Money Do You Need? Part2 (0)