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## Mathematical Foundations: Card Counting Lesson Six Part 1

Mathematics of Card Counting Introduction:

Math explains everything about card counting and the game of blackjack in detail. Because math is a system of absolutes, your “choices” on when to hit or raise your bet shouldn’t involve choices. __A smart card counter is a computer.__ They only do the actions that give them the highest possible mathematical advantage. So far we have had some brief discussions about what advantage means and how mathematical statistics play a role when counting, but in this section we will go into much more detail. This section will be the hardest to figure out, but with a little patience, I guarantee the pieces will fall into place. Because of the possible difficulty, this section has been 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’ll have a full understanding of how math controls your profit and risk.

There are several factors you need to learn to have a full understanding of the math under the cards. We will first discuss the **card counting using expected profit**or – how much you earn from an abstract mathematical point of view. Then we will take blackjack out of the abstract by studying **standard deviation** or – what you actually expect to gain when considering the rules of the game. Once you understand the standard deviation, we can discuss its implications for **risk** and how to reduce this risk to the point where it practically does not exist. Finally, we will discover the possibilities of **exponential profit**.

Expected profit

In short, expected profit is the amount you would earn if your winnings were based solely on the odds you had against the casino. This is the starting point for determining how much cash you will earn.

In mathematical terms, the expected profit for any blackjack hand would be:

A x B = P

Advantage x Bet

=

Expected profit

Since we don’t know in advance exactly which cards you will get, it makes sense to use averages for edge, stake and expected profit. When we talk about average, we use the abbreviation “av”. This changes the expected expected profit formula to:

avA x avB = avP

Avg advantage x avg bet

=

Average expected profit

Of course, you will also be playing more than one hand, so we need to multiply your profit for each hand you play. This gives us the final formula for counting expected profits:

avA x avB x H = avP

Avg Advantage x Avg Bet x Number of Hands

=

Average expected profit

As you can see, increasing any of the left three digits will cause your expected profit to increase. Moreover, because they are multiplied, even small increases will accumulate and lead to much higher profits in the long run. In the beginning, the easiest way to increase your expected profits 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 effect. Let’s say for example that you have played a million hands. If your average bet stays the same at $100, adding half a percent to your edge could realistically mean an expected additional 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, over which you have some control. Let’s go through them one by one starting with the easiest (number of hands) and working your way up to the most complex (medium advantage).

Number of hands played

After perfect counting, most blackjack experts consider this to be the most important factor in making a profit. This number is also the simplest foreword. The more you play, the more you will win. However, there are ways to increase the number of hands you play in addition to playing more hours. The two ways to increase the number of hands you get per hour are:

**1. Faster dealerships** – Casinos and casinos can have a surprising amount in common, and fast dealers are good for both. Because *everyone* who does not count is at a significant disadvantage relative to the house, the expected profit equation becomes an expected loss equation. This means that the more hands the average player gets, the more money the casino makes. So, in general, casinos will do their best to hire and support fast dealers. Is it good news for you.

Obviously, the faster the dealer, the more hands you will get. Most other professional blackjack players will tell you that this is the first thing you need to look for when trying to choose which table to play at, and I totally agree. In the long run, the speed of the dealer will have a huge effect on your bankroll. It will also save you time that you can use for everything from the occasional meal to practicing more blackjack.

There is, however, a downside to a quick dealer. The downside is that a fast dealer can make counting much more difficult (counting becomes even more difficult as more and more people join your table). Dealerships have a pretty simple job, and many of them have been doing it for years. For this reason, some dealers will process at lightning speed. If your counting is not accurate, it can lead to missed counts and totally eliminate any advantage that more hands will give you. When you arrive at a casino, try to sit at the table with the fastest dealer, but if you start to lose count because of their speed, the best thing to do is get up immediately and find a table. slower. If this happens, leave yourself a mental reminder to practice counting bridges as soon as you get the chance.

The fastest dealers are almost always young or middle-aged men. Of course, you shouldn’t assume this will always be true, so be sure to look at a few different dealers to see exactly how fast they work before deciding on a table. Again, a fast dealer is the most important factor to look for once your count is solid at a fast speed.

You should also pay attention to whether or not the casino uses auto mixers. These have made their way into most modern casinos because they allow dealers to spend less time shuffling and more time dealing. They also have drawbacks, but these will only come into play when developing advanced strategies.

**2. Fewer players at your table-** This is one of those things that casinos and counters strongly disagree on. If you’re counting, you want as few people as possible playing at the same table. The fewer people there are, the faster the hands will go. Not only will this increase the number of hands you get, but it will also allow you to increase the number of positive hands, which will also increase your average edge.

The number of people at your table isn’t as manageable as finding a quick dealer, but there are still ways to control this. One thing you can do is try to pick a table where there are either the fewest people or people with the fewest chips on the table. Again, none of these will guarantee that more people won’t sit down or those already at the table won’t buy more, but it will help slightly. Another thing you can do once you’ve already sat down is ask people coming in if they’d wait for the next shoe to start. For some unknown reason, players think it’s a bad idea to go in during a boot. Of course, this doesn’t make sense, but you can use this belief to your advantage, keeping others out so you can fold more hands faster.

__There is also a downside to counting cards with as few people as possible.__. Playing alone can make it much more obvious that you matter. The cameramen will only have you to watch at this table so that your betting habits become more visible. If you feel like you’re feeling hot, I suggest playing 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 likely allow you to play longer, allowing you to have more hands overall.

**Point:** Computer simulations have convinced modern card counters that if you’re not the only person at your table, it’s statistically a good idea to *play two hands at once*. The benefit isn’t huge, but it will definitely add up.

Now that we’ve discussed how to increase the number of hands you can get, let’s throw in some numbers so you can estimate how many hands you’ll get 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 per hour. It would be a counters dream and could lead to sessions where you earn thousands of dollars in an hour or two. In the real world, however, that number is usually a bit lower. A safe overall average is 60 hands per minute, but conditions vary.

If you know the conditions you are facing, go ahead and calculate your own average hands per hour now. Write down this number so that you can have an accurate calculation of your expected profit once we review the other two factors.

In the next part of this lesson, we will discuss the other factors in our calculation of expected profit.

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