Counting and Estimating by Inference
The most obvious unseen card that can be counted with confidence is the dealer's hole card when his up-card is an Ace.
Clearly, if he has not turned up a blackjack and settled everything on the spot, that hole card must be a non-10; thus plus 1 can be incorporated into the temporizing point count during play of a hand. Many players' up-cards are essentially as reliable.
If a player draws a 10 and stands, his hole cards must be xs or he would have broken and turned up his hand. When information is needed for the temporizing count to decide the play of a hand, any hand that has drawn only a 10 and stands counts zero.
Similarly, if a player draws a 10 and another (small) card and stands,his total hand in terms of point count must amount to plus 1 (he has a 10, the x you see, and the two xs you know he must have in the hole).
Occasionally, you may be deceived by a novice who does not know that he must turn up a broken hand. The dealer will quickly instruct such a person in proper play for the purpose of future hands. Also, a player sometimes may misread his hand and stand with a bust.
But these annoyances are uncommon. It is highly advantageous to count unseen hole cards, based upon what reasonably can be inferred from the up-cards that are drawn.
Notice also that when a player draws such a mess of small cards as 2, 3, 4, 4, 2, he must have two more non-20s in the hole; you can see enough cards to total plus 5, but you can infer that the entire hand totals plus 7.
However, no reasonable inference usually can be drawn from a hand that draws a 5 or some other small card and stands. The player may either have two non-10s in the hole or one 10 and one non-10.
In such cases, the unseen cards should be disregarded.
In estimating by inference, it would be possible to devise an entire additional sub strategy of counting by inference, counting on the basis of probable hole cards as derived from the up-cards.
But such a scheme would have to be adapted to the vagaries of play by the incompetents always have to be found at the tables.
It is risky to try to infer the value of hole cards except in obvious situations. Nevertheless, speculative inferences can be made to advantage in some circumstances.
Barring an error by the player, we know, for example, that the draw of a 10 implies two non-10s in the hole if the player stands. But a wealth of information is concealed by 'pat' hands, hands of two hole cards on which the players stand without drawing.
The question arises: can anything be known about these hands? The answer is : not precisely.
But both from mathematics and long experience from a good Twenty-One player have found that much can be estimated: usually a pat hand consists of one 10 and one non-10 (minus 1 total).
Understand that any random two cards should not be assumed to consist of a 1, x. No inference should be drawn about two hole cards unless their holder gives some clue about them. But in the play of Twenty-One, a two-card hand that stands, drawing no cards, usually consists of 10, x.
Some such hands will, of course, be 10,10, and will thus introduce a serious error when counted as minus 1 instead of minus 4; some others will be x, x, and amount to plus 2 instead of the presumed minus 1.
But these errors tend to cancel one another out.
It is better to be approximately right most of the time than to ignore pat hands in taking the temporizing count.
For example, imagine a freshly shuffled deck, and a table with five players. You sit fifth: First hand--- pat; Second hand--- ditto; Third hand--- ditto; fourth hand--- ditto; Your hand, 3, 4; Dealer's hand--- 3 and hole card.
You now draw, and the card is a 5. You have 12. You can see a 3, 4, 3, and 5, for a temporizing count of plus 4. In this strategy, you should stand. But the four pat hands, simply because they are pat, very probably contain several 10s.
By the criteria,you have just given for an estimated temporizing count, each is counted minus 1, and the temporizing count, each is counted minus 1, and the temporizing count reverts zero. Accordingly, you should draw.
At the showdown, the cards may well be as follows: First hand--- 10, 7 (minus 1); Second hand--- 10, 10 (minus 4); Third hand--- 9, 8 (plus 2); Fourth hand--- 10, 6 (minus 1); Your hand--- 3, 4, 5, 5; Dealer's hand--- 3, 10, 4.
You tie instead of losing. The sample hands, of course, are invented to make a point. Had all the 10s in the hidden pat hands been ignored, you would not have dared to draw to your 12, because of the inaccurate plus 4 temporizing count.
But the estimated zero count (plus 4 and minus 4 equal zero) induced you to draw your expected small card and salvage your bet. The temporizing count is a provisional count--- one to be corrected when all available data are known but to be put to practical use when needed in the interim.