What Dealer Upcard Is Best for Blackjack Players?

The majority of solid citizens would guess that a five or a six is the most advantageous dealer upcard for blackjack players. Depending on rules in force at a table and, more significantly, what's in your own hand, either or neither may actually be best.

The number of decks in play has an influence. For instance, in a one-deck game where the dealer stands on soft 17, five-up is expected to break eight times more often per thousand tries than six-up; in the same game with six decks, six-up is expected to bust four times more often per thousand. A difference, but small.

There are, however, common cases where the issue is relevant and distinctions among alternative upcards are large. For instance, assume a multi-deck game where the dealer stands on soft 17, and you're sophisticated enough to follow Basic Strategy.

Say you start with 9-5, hard 14. You're an underdog when you get this total, regardless of the dealer's upcard. The accompanying list shows how much you "expect" to lose for every $10 bet if the round is playable (the dealer doesn't have a blackjack). The figures confirm that the least of the evils is a six?up, when you'd stand and the dealer is most apt to bust.

expectations when starting with 9-5, for every $10 bet
upcard	action	expectation
two	stand	lose $2.90
three	stand	lose 2.51
four	stand	lose 2.08
five	stand	lose 1.63
six	stand	lose 1.53
seven	hit	lose 3.24
eight	hit	lose 3.72
nine	hit	lose 4.30
ten	hit	lose 4.65
ace	hit	lose 4.40

Pretend, instead, you start with 10-8. Here, your prospects are middling. Hard 18 has a positive expectation with seven of the 13 dealer upcards and negative with the other six, as shown in the next list. Notice that, standing on 10-8, the most propitious condition for you is not when the dealer has six-up and is likeliest to crash. It's against seven. The reason is that you'll win with your 18 if the dealer breaks or finishes with 17, push if the dealer ends with 18, and lose only against 19 through 21.

expectations when starting with 10-8, for every $10 bet
upcard	action	expectation
two	stand	win $1.21
three	stand	win 1.48
four	stand	win 1.74
five	stand	win 2.00
six	stand	win 2.82
seven	stand	win 3.98
eight	stand	win 1.04
nine	stand	lose 1.85
ten	stand	lose 1.75
ace	stand	lose 0.98

What are the probabilities of winning and losing that underlie the expectations for 10-8 being stronger against seven than six?

With six-up, the dealer has 42.3 percent chance of busting and 16.5 percent of a final 17, giving you a 58.8 percent shot at success. The probability that the dealer will end with 18, for a push, is 10.6 percent. And you'll fail if the dealer goes to 19, 20, or 21 - a combined unfavorable outlook of 30.6 percent. If you're mathematically inclined, you'll see that the $2.82 expected profit on the $10 bet comes by taking 58.8 percent of the $10 you can win and subtracting 30.6 percent of the $10 you can lose. This works out to be $5.88 minus $3.06 or $2.82.

With seven-up, the dealer has only 26.2 percent chance of going belly-up, but a healthy 36.8 percent likelihood of stopping on 17. The other probabilities are 13.8 percent of 18 and 23.2 percent of 19 through 21. So your prospects are 26.2 + 36.8 = 63.0 percent ecstasy, 13.8 percent neutral, and 23.2 percent agony. Again, working this out for a $10 bet, you get $6.30 minus $2.32 for the $3.98 expected profit shown in the table.

There's lots of talk about "junk science" these days. The worst is pure subterfuge. Some is simply ingenuous, based on the idea that knowing selected facts helps get around them. Good science means understanding how to use data most effectively. So it is with gambling. Anyone who believes otherwise must have ignored this injunction of the inimitable inkslinger, Sumner A Ingmark:

The wise approach with apprehension,
Naive attempts at circumvention.