A distinctive ability of a good player is the ability to count cards that have left the game. Such players are called counters for a reason. When playing blackjack, they carefully monitor the cards used, trying to adjust the probability of the next cards appearing and predict the course of further events. However, we will start by looking at a very simple model based on the fact that cards of different denominations come out of the deck evenly. In other words, we assume that the probability of the appearance of a card of a certain rank ( say, an ace or an eight) is 1/13, tens – 4/13 ( recall that in addition to the usual tens, this includes pictures).

The hypothesis we have adopted reflects the simple case when the players do not make special calculations, but act only according to common sense, believing that ” distortions” are rare. On the other hand, you can imagine an endless deck, consisting of an unlimited number of 52-card decks, or the usual ” big deck”, where all the cards not participating in the current deal are constantly mixed and shuffled. In such situations, card counting would be meaningless, and our hypothesis would become an immutable fact.

However, this is not so far away. At one time, a lot of counters appeared in the casino, which put the existence of blackjack at risk, but the antidote was found pretty quickly. Third, or even half of the ” big deck” were excluded from the game with a special card ( cutting card), and the effectiveness of card counting significantly decreased. And in many European casinos, mechanisms have already appeared that at any time allow the dealer to mix the played cards with the remaining part of the deck in the game. The cards are shuffled randomly, which effectively means shuffling. Since the cards that are out of the game are constantly being returned to the deck, the shuffle becomes potentially infinite …

Now we will consider a strategy of the game, which, within the framework of the uniformity hypothesis, allows the player to act optimally. But since the strategy works with this concept in mind, it is called not optimal, but basic. Indeed, counters trying to assess the current situation at the table will find an excellent starting point in it. We’ll come back to the issue of card counting and adjusting the basic strategy later.

## ADDITIONAL CARD SET

So you see your two initial cards and the dealer’s starting card. Sooner or later, he will start to collect his card combination and end up getting from 17 to 21 points or running into a bust. The probabilities of different outcomes depending on the dealer’s card are shown in Table 1.

Obviously, the dealer’s starting cards are not equal. The first ( and fairly accurate) idea of the strength of his card can be obtained from the ” Overkill” column . The strongest card is an ace, the weakest is 6. And if you line up the cards in decreasing order of strength, we get the sequence

T, 10, 9, 8, 7, 2, 3, 4, 5, 6,

although a more accurate analysis leads to a slightly different result:

T, 10, 9, 8, 2, 3, 7, 4, 5, 6.

Now let’s look at the player’s cards and try to understand whether it is worth taking an additional card or is it better to stop. If blackjack comes, there is nothing to talk about, if 19 or 20 points is also not bad, and a new card is not needed. Although it is easy to deduce from Table 1 that the dealer’s chances of getting an ace or ten are slightly higher than that of a player with 19 points.

Let’s assume for now that you don’t have an ace: the double-valued aces ( 1 or 11 points) requires a separate approach. It can be proved that it is not profitable to take a card with 17 or 18 points. On the contrary, if you have 11 points or less, you must take a card. This is quite obvious: you are insured against busting, and a new card will not reduce points.

12-16 points is a difficult situation: your chances are slim, and your strategy should depend on the dealer’s card. Let’s say he has an ace and you have 16 points. Only in 5 cases out of 13 a new card ( 2, 3, 4, 5 or ace) will improve your combination, in the remaining 8 cases you will be bust ( 6, 7, 8, 9 and four tens). Therefore, it seems that taking the card is insane. But this is not so: the dealer’s card is very strong and the situation is so difficult that the risk is simply necessary. This typical ” gambling” reasoning is confirmed, of course, by mathematics. If you give up the extra card, you will lose ( on average) 76.94 cents on every dollar, and the “ risky” decision will reduce the loss to 66.57 cents. The stronger the dealer’s card, the more justified the risk.

The strategy of an additional set of cards is shown in Table 2 ( “+” – take a card, ” -” – stop). If at least one of the player’s two initial cards is an ace, we use Table 3. It is essential that the set of all subsequent cards is also based on these two tables, corresponding to hard and soft combinations. Sometimes tables have to be used interchangeably. The fact is that hard combinations can turn into soft ones, and vice versa. Let’s look at some examples.

Your cards are 4-3, the dealer has a six. We have 7 ” hard” points, we look at table 2 and make a decision: take an additional card. Let’s say it’s an ace. A hard combination turned into a soft one: 8 or 18 points on the hands. Table 3 shows that when the dealer has a six, dialing should be stopped;

Your cards are T-4, the dealer has a nine. Having 5 or 15 ” soft” points, we use table 3 and take an additional card. Let it be a two, i.e. we have 7 or 17 points. Referring again to table 3 and get a new card. If it is 10, then there are 17 ” hard” points on hand , and table 2 will show that it is time to stop. But if it is an ace, we have 8 or 18 points and according to table 3 we take the card again.

## SPLIT CARD, DOUBLE BET, FAIL TO PLAY

Having received the original two-card hand, first of all, you need to investigate it for splitting ( or, as the players say, splitting). If your two cards are of the same rank, use table 4. It shows when you need to split cards ( Spl), and in what cases it is unprofitable ( empty cells of the table). We emphasize that Table 4 has the highest priority and is the starting point for analyzing the situation. Sometimes, however, you will have to face the limitation on the maximum number of splits allowed in the casino ( no more than four combinations on one box). When separation is no longer possible, it is naturally ignored in our analysis.

The next step – the possibility to double check rates ( double) and the rejection of the game – ” compensation» ( the surrender). The strategy of doubling bets on hard and soft combinations is shown in tables 5, 6 ( Dbl – doubling). It can be seen from the tables that doubling is unprofitable on ” hard” glasses going out of the range of 9-11, although the spectrum of possibilities is somewhat wider on soft combinations.

As for the refusal or ” compensation”, then this technique is advisable only on hard combinations, and even then in rather rare cases listed in Table 7 ( Sur – refusal). Most often, “ buy-off” makes sense if the dealer’s starting card is an ace.

And, finally, if doubling the bet or “ payoffs” turned out to be meaningless, we turn to tables 2, 3 and move on to a set of additional cards.

Please note that due to blackjack rules, tables 4-7 only apply to the original two-card hands. Tables 4-6 can be used for new two-card combinations resulting from splits, but table 7 does not apply in these cases. For combinations with three or more cards, all information is concentrated in tables 2, 3.

We reviewed the basic strategy of the game of blackjack, excluding insurance ( insurance), which will be discussed below. The abundance of tables can cause inconvenience, so all the necessary information is collected in tables 8 ( for hard combinations) and 9 ( for soft combinations). In each cell of both tables there is a sign of the continuation of the set of additional cards ( “+” or ” -“). The rest of the designations are clear without comment.