How Well Can You Predict Dealers' Final Totals from their Upcards?

25 October 2006

By Alan Krigman

When blackjack buffs see dealer upcards of seven, they often think "17" to themselves. Similarly for eight-up, "18." For nine, "19." For 10, "20." And for ace -- well, considering the family fun of playing kitchen-table blackjack for buttons, I won't say.

What about upcards of six and under? Most players confidently, maybe overconfidently, think "oh-boy." Except possibly on two-up when at least the cognoscenti have been heard to murmur "oh-oh."

In fact, prospects of sevens maturing to 17s, eights to 18s, and so on are strong but by no means certain. And, while upcards from two through six are rightly regarded as "stiffs," the chances of their leading to busts are less than bettors typically believe. Further, low upcards vary widely in the probabilities of the totals they reach when the hands don't exceed 21.

For simplicity, picture an infinite shoe. That is, the proportions of cards remaining to be drawn are always one out of 13 for each rank other than 10 and four out of 13 for 10-values. The numbers won't be far off for finite six or eight deckers. Then, the first of the accompanying tables gives probabilities of all possible dealer finishes starting with seven through ace.

The data for seven- through ace-up do indicate that the most likely resolution is the face value plus 10. However, other than with ace-up, a bust is second most probable. And, totals lower than face value plus 10 are expected more often than higher.

What about upcards from two through six? Probabilities for these situations are given in the second of the accompanying tables.

Probabilities of alternate dealer totals for upcards from
two through six

With two- through six-up, busting is indeed the most likely outcome. However, the chance of this result is always shy of 50 percent; it's least at 34.36 percent with a two and most at 42.31 percent with a six. Therefore, when solid citizens stand against these upcards with totals below 16, they're more likely to lose to a dealer's total between 17 and 21 than win to a bust.

Standing at or below 16 against stiffs, other than with 12 versus two- or three-up, doesn't make players the favorites. It's the preferred option because it loses less often, on the average, than hitting. Either because hitting busts and loses however the dealer fares, or yields a total the dealer exceeds anyway.

The situation differs with 12 versus two or three. Basic Strategy is to hit under these circumstances. The reason is twofold.

First, of the low upcards, the table shows dealers busting least often with twos and threes. This heightens the danger of sitting there with a total vulnerable to any "made" hand. Second, the chance of reaching 17 through 21 is five out of 13 (38.5 percent) regardless of a player's two-card total of 16 or below. However, starting with 12 or 13, the probability of drawing to a bust is four or five out of 13 (30.8 to 38.5 percent), respectively. With a two-card 14 through 16, it rises to a much worse six through eight out of 13 (46.2 to 61.5 percent).

All of which points to one of the key reasons blackjack has been a perennial favorite. And also illustrates why many weak players stand when the math says to hit. A round ain't over 'til it's over. Or, as that perspicacious poet, Sumner A Ingmark, put it:

Excitement builds as tabulation,
Increases the anticipation;
So in the normal course of play,
An underdog can have its day.

rnn