Mathematics of Card Counting Introduction:
Mathematics explains in clear detail everything about card counting and the game of blackjack. Because mathematics is a system of absolutes, your "choices" about when to hit or raise your bet should not actually involve choice. A smart card counter is a computer. They make only the actions that grant them the highest possible mathematical advantage. So far we have had some brief discussions regarding what advantage means and how mathematical statistics play a role when counting, but in this section we will go into much greater detail. This section will be the most difficult to understand, but with a little patients I guarantee the pieces will fall into place. Because of the possible difficulty, this section was written with the average reader in mind. Even if you haven't taken a math class since high school, by the end of this lesson you will have a complete understanding of how mathematics controls your profit and your risk.
There are several factors you must learn to have a complete understanding of the math beneath the cards. First we will discuss the card counting use of expected profitor - how much you are winning from an abstract mathematical point of view. Then we will take blackjack out of the abstract by studying standard deviation or - what you will actually expect to be winning when you account for the rules of the game. Once you understand standard deviation, we can discuss its implications for risk and how to lower this risk to the point where it essentially does not exist. Finally we will learn about opportunities for exponential profit.
Expected Profit
In short, Expected profit is how much you would win, if your winnings were only based on the odds you had against the casino. This is the starting point for figuring out how much money you will make counting.
In mathematical terms expected profit for any single blackjack hand would be as follows:
A x B = P
Advantage x Bet
=
Expected profit
Since we do not know ahead of time exactly what cards you will get, it makes sense to use averages for advantage, bet, and expected profit. When discussing average lets use the abbreviation "av". This changes the expected expected profit formula to:
avA x avB = avP
Avg advantage x Avg bet
=
Avg Expected Profit
Of course you will also be playing more than one hand, so we have to multiply your profit for every hand you play. This gives us the final formula for card counting expected profit:
avA x avB x H = avP
Avg Advantage x Avg Bet x Number of Hands
=
Avg Expected Profit
As you can see, increasing any one of the three numbers on the left will lead to an increase in your expected profit. Not only that, because they are multiplied, even small increases will stack up and lead to much higher profits in the long run. At first the easiest way to increase your expected profit will be to play more hands. For example, playing 2000 hands instead of 1000 hands will double your expected profit. Over time however, the other two numbers will have more and more of an effect. Lets say for example that you have played one million hands. If your average bet stays the same at $100, then adding half of one percent to your advantage could realistically mean an added expected profit of over five hundred thousand dollars.
So, now that we have a formula for expected profit, what number should you use in place of each letter? Well it depends on several factors, most of which you have some control over. Lets talk through them one at a time starting from the easiest (number of hands) and working our way to the most complex (average advantage).
Number of Hands Played
After perfect counting, most blackjack experts consider this the most important factor to making profit. This number is also the most straight foreword. The more you play the more you are going to make. However, there are ways to increase the number of hands you play besides playing more hours. The two ways to increase how many hands you get in per hour are as follows:
1. Faster Dealers - Casinos and counters can have a surprising amount in common, and fast dealers are good for both of them. Because everyone that is not counting is at a solid disadvantage against the house, the expected profit equation becomes an expected loss equation. This means the more hands the average gambler gets in, the more money the casino makes. So in general, casinos will try their best to hire and support fast dealers. This is good news for you.
Obviously the faster the dealer is, the more hands you will get in. Most other professional blackjack players will tell you that this is the first thing you should look for when trying to pick which table to play at, and I totally agree. Over the long run the speed of the dealer will have a huge effect on your bankroll. It will also save you time which you can use for everything from the occasional meal to practicing more blackjack.
There is however a disadvantage to a fast dealer. The disadvantage is that a fast dealer can make it much harder to count (counting becomes even more difficult as more people join your table). Dealers have a pretty simple job, and many of them have been doing it for years. Because of this some dealers will deal at a lightning pace. If your counting is not razor sharp this can lead to missed counts and totally remove any advantage getting in more hands will give you. When you get to a casino, try and sit down at the table with the fastest dealer, but if you start losing the count because of their speed, the best thing you can do is get up immediately and find a slower table. If this happens, leave yourself a mental reminder to practice counting down decks as soon as you have the chance.
The fastest dealers are almost always young to middle aged men. Of course you should not assume that this will always be true, so make sure to watch a few different dealers to see exactly how fast they work before picking a table. Again, a fast dealer is the single most important factor to look for once your counting is solid at a lickidy speed.
You should also pay attention to whether or not the casino uses automatic shufflers. These have made their way into most modern casinos as they allow dealers to spend less time shuffling and more time dealing. There are disadvantages to them as well, but these will only come into play when preforming advanced strategies.
2. Less players at your table- This is one of those things casinos and counters strongly disagree about. If you are counting, you want as few people playing at the same table as you can. The less people there are the quicker hands will go. Not only will this increase the number of hands you get in, but it will also allow you more hands in positive counts which effectively raises your average advantage as well.
The number of people at your table is not as manageable as finding a quick dealer but there are ways of controlling this all the same. One thing you can do is to try and pick a table where there are either the fewest people, or the people who have the smallest amount of chips on the table. Again neither of these will guarantee that more people don't sit down or that those already at the table won't buy-in for more, but it will help slightly. Another thing you can do once you have already sat down is to ask people coming in if they would wait for the next shoe to start. For some unknown reason gamblers think it is a bad idea to come in during a shoe. This of course makes no sense whatsoever but you can use this belief to your advantage, keeping others out so you can get through more hands more quickly.
There is also one downside to card counting with as few people as possible. Playing by yourself can make it much more obvious that you are counting. The camera operators will only have you to watch at that table so your betting patterns become more visible. If you feel like you've been getting heat I would suggest playing at tables with two or three other players. This won't have a huge effect on the number of hands you can play per shoe and will probably allow you to play for longer stretches of time, allowing you to get more hands in over all.
Tip: Computer simulations have convinced modern card counters that if you are not the only person at your table, it is statistically a good idea to play two hands at a time. The advantage is not huge, but it will certainly add up.
Now that we have discussed how to increase the number of hands you are able to get, lets throw out some numbers so you can estimate how many hands you will be getting per hour based on your conditions. In an ideal situation (a fast dealer with no other players) you can realistically hit between 120 and 140 hands an hour. This would be a counters dream and could lead to sessions where you are making thousands of dollars in an hour or two. In the real world however this number is usually a bit lower. A safe overall average is 60 hands per minute, but conditions will vary.
If you know the conditions you are dealing with then go ahead and calculate your own average hands per hour now. Write this number down so you can have an accurate calculation of your expected profit once we go through the other two factors.
In the next part of this lesson we will discuss the other factors in our expected Profit calculation.
Before going to the casino, learn more about card counting. Everything a card counter needs is in the book Modern Card Counting: Modern Card Counting. This ebook includes lessons on everything from basic strategy to using mathematical formula to estimate your profit and expected risk.
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