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Statistics Hacks by Bruce Frey

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Hack #40. Play in the Black in Blackjack

Perhaps the most potentially profitable application of statistics hacking is at the blackjack table.

In blackjack, the object of the game is to get a hand of cards that is closer to totaling 21 points (without going over) than the dealer's cards. It's a simple game, really. You start with two cards and can ask for as many more as you would like. Cards are worth their face value, with the exception of face cards, which are worth 10 points, and Aces, which can be either 1 or 11.

You lose if you go over 21 or if the dealer is closer than you (without going over). The bets are even money, with the exception of getting a blackjack: two cards that add up to 21. Typically, you get paid 3-to-2 for hitting a blackjack. The dealer has an advantage in that she doesn't have to act until after you. If you bust (go over 21), she wins automatically.

Statisticians can play this game wisely by using two sources of information: the dealer's face-up card and the knowledge of cards previously dealt. Basic strategies based on probability will let smart players play almost even against the house without having to pay much attention or learn complicated systems. Methods of taking into account previously dealt cards are collectively called counting cards, and using these methods allows players to have a statistical advantage over the house.

Tip

U.S. courts have ruled that card counting is legal in casinos, though casinos wish you would not do it. If they decide that you are counting cards, they might ask you to leave that game and play some other game, or they might ban you from the casino entirely. It is their right to do this.

Basic Strategy

First things first. Table 4-12 presents the proper basic blackjack play, depending on the two-card hand you are dealt and the dealer's up card. Most casinos allow you to split your hand (take a pair and split it into two different hands) and double down (double your bet in exchange for the limitation of receiving just one more card). Whether you should stay, take a card, split, or double down depends on the likelihood that you will improve or hurt your hand and the likelihood that the dealer will bust.

Table 4-12. Basic blackjack strategy against dealer's up card

Your handHitStayDouble downSplit
5-8Always   
92, 7-A 3-6 
10-1110 or A 2-9 
122, 3, 7-A4-6  
13-167-A2-6  
17-20 Always  
2, 28-A  2-7
3, 32, 8-A  3-7
4, 42-5, 7-A  6
5, 510 or A 2-9 
6, 67-A  2-6
7, 78-A  2-7
8, 8   Always
9, 92-6, 8, 9  7, 10, A
10, 10 Always  
A, A   Always
A, 22-5, 7-A 6 
A, 3 or A, 42-4, 7-A 5 or 6 
A, 52 or 3, 7-A 4-6 
A, 62, 7-A 3-6 
A, 79-A2, 7-A3-6 
A, 8 or 9 or 10 Always  

Tip

In Table 4-12, "Your hand" is the two-card hand you have been dealt. For example, "5-8" means your two cards total to a 5, 6, 7, or 8. "A" means Ace. A blank table cell indicates that you should never choose this option, or, in the case of splitting, that it is not even allowed.

The remaining four columns present the typical options and what the dealer's card should be for you to choose each option. As you can see, for most hands there are only a couple of options that make any statistical sense to choose. The table shows the best move, but not all casinos allow you to double-down on just any hand. Most, however, allow you to split any matching pair of cards.

Why It Works

The probabilities associated with the decisions in Table 4-12 are generated from a few central rules:

  • The dealer is required to hit until she makes it to 17 or higher.

  • If you bust, you lose.

  • If the dealer busts and you have not, you win.

The primary strategy, then, is to not risk busting if the dealer is likely to bust. Conversely, if the dealer is likely to have a nice hand, such as 20, you should try to improve your hand. The option that gives you the greatest chance of winning is the one indicated in Table 4-12.

Tip

The recommendations presented here are based on a variety of commonly available tables that have calculated the probabilities of certain outcomes occurring. The statistics have either been generated mathematically or have been produced by simulating millions of blackjack hands with a computer.

Here's a simple example of how the probabilities battle each other when the dealer has a 6 showing. The dealer could have a 10 down. This is actually the most likely possibility, since face cards count as 10. If there is a 10 down, great, because if the dealer starts with a 16, she will bust about 62 percent of the time (as will you if you hit a 16).

Since eight different cards will bust a 16 (6, 7, 8, 9, 10, Jack, Queen, and King), the calculations look like this:

8/13 = .616

Of course, even though the single best guess is that the dealer has a 10 down, there is actually a better chance that the dealer does not have a 10 down. All the other possibilities (9/13) add up to more than the chances of a 10 (4/13).

Any card other than an Ace will result in the dealer hitting. And the chances of that next card breaking the dealer depends on the probabilities associated with the starting hand the dealer actually has. Put it all together and the dealer does not have a 62 percent chance of busting with a 6 showing. The actual frequency with which a dealer busts with a 6 showing is closer to 42 percent, meaning there is a 58 percent chance she will not bust.

Now, imagine that you have a 16 against the dealer's down card of 6. Your chance of busting when you take a card is 62 percent. Compare that 62 percent chance of an immediate loss to the dealer's chance of beating a 16, which is 58 percent. Because there is a greater chance that you will lose by hitting than that you will lose by not hitting (62 is greater than 58), you should stay against the 6, as Table 4-12 indicates.

All the branching possibilities for all the different permutations of starting hands versus dealers' up cards result in the recommendations in Table 4-12.

Simple Card-Counting Methods

The basic strategies described earlier in this hack assume that you have no idea what cards still remain in the deck. They assume that the original distribution of cards still remains for a single deck, or six decks, or whatever number of decks is used in a particular game. The moment any cards have been dealt, however, the actual odds change, and, if you know the new odds, you might choose different options for how you play your hand.

Elaborate and very sound (statistically speaking) methods exist for keeping track of cards previously dealt. If you are serious about learning these techniques and dedicating yourself to the life of a card counter, more power to you. I don't have the space to offer a complete, comprehensive system here, though. For the rest of us, who would like to dabble a bit in ways to increase our odds, there are a few counting procedures that will improve your chances without you having to work particularly hard or memorize many charts and tables.

The basic method for improving your chances against the casino is to increase your wager when there is a better chance of winning. The wager must be placed before you get to see your cards, so you need to know ahead of time when your odds have improved. The following three methods for knowing when to increase your bet are presented in order of complexity.

Counting Aces

You get even money for all wins, except when you are dealt a blackjack. You get a 3-to-2 payout (e.g., $15 for every $10 bet) when a blackjack comes your way. Consequently, when there is a better-than-average chance of getting a blackjack, you would like to have a larger-than-average wager on the line.

The chances of getting a blackjack, all things being equal, is calculated by summing two probabilities:

Getting a 10-card first and then an Ace

4/13x4/51 = .0241

Getting an Ace first and then a 10-card

1/13x16/51 = .0241

Add the two probabilities together, and you get a .0482 (about 5 percent) probability of being dealt a natural 21.

Obviously, you can't get a blackjack unless there are Aces in the deck. When they are gone, you have no chance for a blackjack. When there are relatively few of them, you have less than the normal chance of a blackjack. With one deck, a previously dealt Ace lowers your chances of hitting a blackjack to .0362 (about 3.6 percent). Dealing a quarter of the deck with no Aces showing up increases your chances of a blackjack to about 6.5 percent.

Tip

Quick tip for the budding card counter: don't move your lips.

Counting Aces and 10s

Of course, just as you need an Ace to hit a blackjack, you also need a 10-card, such as a 10, Jack, Queen, or King. While you are counting Aces, you could also count how many 10-cards go by.

There is a total of 20 Aces and 10-cards, which is about 38 percent of the total number of cards. When half the deck is gone, half of those cards should have been shown. If fewer than 10 of these key cards have been dealt, your chances of a blackjack have increased. With all 20 still remaining halfway through a deck, your chances of seeing a blackjack in front of you skyrockets to 19.7 percent.

Going by the point system

Because you want proportionately more high cards and proportionately fewer low cards when you play, a simple point system can be used to keep a running "count" of the deck or decks. This requires more mental energy and concentration than simply counting Aces or counting Aces, 10s, and face cards, but it provides a more precise index of when a deck is loaded with those magic high cards.

Table 4-13 shows the point value of each card in a deck under this point system.

Table 4-13. Simple card-counting point system

CardPoint value
10, Jack, Queen, King, Ace-1
7, 8, 90
2, 3, 4, 5, 6+1

A new deck begins with a count of 0, because there are an equal number of -1 cards and +1 cards dealt in the deck. Seeing high cards is bad, because your chances of blackjacks have dropped, so you lose a point in your count. Spotting low cards is good, because there are now proportionately more high cards in the deck, so you gain a point there.

Tip

You can learn to count more quickly and easily by learning to rapidly recognize the total point value of common pairs of cards. Pairs of cards with both a high card and a low card cancel each other out, so you can quickly process and ignore those sorts of hands. Pairs that are low-low are worth big points (2), and pairs that are high-high are trouble, meaning you can subtract 2 points for each of these disappointing combinations.

You will only occasionally see runs of cards that dramatically change the count in the good direction. The count seldom gets very far from 0. For example, with a single new deck, the first six cards will be low less than 1 percent of the time, and the first ten cards will be low about 1/1000 of 1 percent of the time.

The count doesn't have to be very high, though, to improve your odds enough to surpass the almost even chance you have just following basic strategy. With one deck, counts of +2 are large enough to meaningfully improve your chances of winning. With more than one deck, divide your count by the number of decks—this is a good estimation of the true count.

Sometimes you will see very high counts, even with single decks. When you see that sort of string of luck, don't hesitate to raise your bet. If you get very comfortable with the point system and have read more about such systems, you can even begin to change the decisions you make when hitting or standing or splitting or doubling down.

Even if you just use these simple systems, you will improve your chances of winning money at the blackjack tables. Remember, though, that even with these sorts of systems, there are other pitfalls awaiting you in the casino, so be sure to always follow other good gambling advice [Hack #35] as well.

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