15 v 10

Should I hit an 8-7 fifteen against a dealer’s ten?

It depends on the kind of game you’re playing.

The master tables shown in the Basic Strategy section of this site indicate that you should surrender fifteen—any fifteen—against a dealer’s ten-value card, but these decks are based upon a multiple-deck game in which the dealer stands on a soft 17.

There are additional tables under Situational Strategy that indicate a different decision when playing a single-deck game: hit rather than surrender on a fifteen—again, any fifteen—against a ten.

The reason for this difference is the effect of the removal of cards from the deck.

In a multiple-deck game, the removal of a card has a negligible effect. In a six-deck shoe, removing a nine means that there are 23 remaining nines in the 311 remaining unseen cards. This means there’s a 7.39% chance of seeing another nine, as opposed to the 7.69% (24 nines among 312 unknown cards) chance of drawing a nine before a nine was seen. The difference, 0.3%, is statistically insignificant.

In a single-deck game, however, the odds of seeing the second nine decrease from 7.69% (four nines in 52 cards) to 5.88% (three in 51) once a nine has been seen, so the difference (1.81%) is much more dramatic, and worth considering.

The removal of high cards (anything over a six) makes it less likely your fifteen will bust on the next hit. If you have an eight-seven fifteen, two cards that can bust you (the eight and seven you’ve already been dealt) are no longer in the deck, making it 3.62% less likely you’ll bust … whereas it would be only 0.60% less likely if it were a multiple-deck game.

It’s worth noting that the single-deck strategy table is an aggregate—the advice to hit a fifteen is based on the odds of all three species of hard fifteen: the eight-seven, six-nine, and ten-five. Because the latter two hands consume only one bad hit (the 9 or 10 would bust you), and at the same time consume a good hit (the 5 or 6 would bring the hand to 20 or 21), the two generally cancel one another out.

If you were to stand on a hard 15, you’d win only 24.98% of the time (when the dealer busts), which is plainly worse than the odds that warrant surrendering (25.00%). If you hit a six-nine or five-ten hand, you will win 24.95% of the time, which is slightly worse (0.03%) than if you merely stood pat—but still below the odds where surrender is warranted. If you hit an eight-seven fifteen, you would will 25.05% of the time, which is slightly better (0.07%) than standing pat and, at the same time, slightly better than the odds where surrender is warranted.

From a purely mathematical standpoint, a player who hits instead of surrenders his eight-seven fifteens in a single-deck game will increase his bankroll by 0.005%—but to put it into perspective, that’s a nickel more for every $1,000 in action (40 hands at $25, 100 hands at $10, or 200 hands at $5).

Deciding whether memorizing this specific situation is worth that nickel-per-thousand rewardis entirely up to you—but the more often you play and the more money you wager on eachhand, the more valuable this special case becomes.