So Much for Assuming that the Unseen Card is a 10

15 June 2005

By Alan Krigman

A theory held and acted upon by far too many blackjack buffs, not all of whom are babes in the woods, is that in a pinch it's safe to assume "the unseen card is a 10." This says, for instance, a dealer's 10-up is headed toward a total of 20. And similarly, only an extreme stroke of luck will keep a dealer who starts with a six from busting, since the presumed 10 in the hole will be followed by a like draw, sending the sum to La-La-Land. Among other things, this lame logic leads legions of bettors to wonder whether "the book" may conceivably be wrong on certain admittedly close calls. One such situation is the edict that a total of 12 be hit against a 2-up. Another is the dictum that a pair of eights be split when the dealer has a 10, and the halves played individually with twice as much moolah at risk. A standard deck has four 10-valued ranks and nine others. A 10 will therefore appear more often than anything else. Ignoring finite shoe size and cards already withdrawn, including those on the table, each rank from ace through nine is expected 1/13 or 7.69 percent of the time. A 10 is projected to pop the remaining 4/13 or 30.77 percent. If you want to believe that 30.77 percent is a guarantee, it's your dough. Maybe not for long, though. Of course, a 10-up can end at 20 with other than a 10 in the hole. Flipping a six and pulling a four, or a three and a seven, are among the possibilities. And, what of dealers' chances to reach other points, including a bust? Dealer results for 2- and 10-up, again ignoring shoe size and withdrawal effects, are shown in the accompanying table. In both instances 10s have a strong effect compared with other "unseen" cards. Strong but not overwhelming. Overall, the data can help you understand what's behind the 12 versus two and eight-eight against 10 enigmas. Probability of dealers' final results starting with upcards of 10 or 2 Finish 2-up 10-up 17 13.98% 11.14% 18 13.49% 11.14% 19 12.96% 11.14% 20 12.40% 34.22% 21 11.80% 3.45% BJ -- 7.69% Bust 35.36% 21.21% With 2-up, a dealer's bust is the most probable outcome. But 35.36 percent is still just over one third. Other results range from nearly 12 to almost 14 percent, with a bias toward the low totals. Standing on 12 and hoping the dealer will break therefore has 35.36 percent chance of winning. Hitting the 12 will put you above 21 and out of action 4/13 or 30.77 percent of the time, and is no worse or offers a shot at beating the dealer the remaining 68.23 percent. When the dust has settled on the decimal points, it's a close call. You're an underdog either way, but average expected losses are lower hitting under these circumstances. Starting with 10-up, a 20 is the most likely dealer result, although it's only anticipated in slightly more than a third of all these hands. And, you may be surprised that the next most likely result is a bust, in over a fifth of all cases. Based purely on what can happen to a 16 or separate eights, statistical analysis shows splitting the pair as the preferred option. The fly gets into the ointment with eights versus 10, for two reasons. First, you've got a handicap no matter what you do, so the issue boils down to the least of the evils. Second, splitting the pair to cut the disadvantage means solid citizens must match the money already at risk and face the danger of losing the whole amount. The predicament is a classic gambling conundrum. The laws of probability point toward splitting. But utility theory suggests that less cash up for grabs now may be preferable to a lower penalty averaged out later. A similar dilemma confronts blackjack players with hands appropriate for doubling down. Some balk at the higher bet, despite having favorable odds. It's a matter of the promise of profit versus the potential penalty. Not on an absolute objective scale, but in how an individual views the pleasure and the pain of the possible results. And who's to argue personal preference? Surely not the sagacious songster, Sumner A Ingmark, who said: When of dollars there's an awful dearth, Even fools should know what money's worth.