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Best of Alan Krigman

Gaming Guru

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Playing It Smart: Can you gain an edge in blackjack by counting only aces?

14 April 2009

An oddity of blackjack is that cards may be favorable in some situations and adverse in others. Card counters know that shoes rich at the high end help them while those with excess low ranks serve the house, so they raise or lower their bets accordingly.

But, consider this. Say you're dealt a 5-6 and double-down. You'd like to pull a 10. However, you need those fives and sixes to get into this catbird seat. And, if you pulled 6-6 against a three, you'd like to split the pair in which case a 10 on one or both wouldn't knock you out but would certainly leave you vulnerable.

Regardless, solid citizens generally like to see aces coming at them. One reason is that pairs of aces, when split, tend to be profitable. Another is that aces offer a second chance in soft hands; for instance, pulling a five or six to A-4 would be great but seven or above leaves a new shot at drawing successfully.

The most evident benefit of aces, arguably the most important, is their role in blackjacks. Everyone likes to get blackjacks. The worst they do is push if the dealer also lucks out. Otherwise, they win automatically and also pay 3-to-2 a half-bet bonus.

An indirect boon of this bonus is its impact on the house advantage. Were blackjacks to pay even money, they'd be coveted anyway. But they'd have no impact on edge because players and dealers have equal prospects of getting them.

With bettors winning 3-to-2 while losing only what's at risk, blackjacks in and of themselves cut the inherent edge in the game by one-half (the half-bet bonus) times the probability of getting the hand. In shoes with the original proportions of cards, this amounts to a sizeable 2.373, 2.374, and 2.238 percent in eight-, six-, and four-deck games, respectively.

As the shoe is depopulated, the proportions of each rank change and the edge varies accordingly. To envision the impact, make believe you monitor the number of aces discarded and assume that 10-valued cards remain in their original proportions. The accompanying table shows representative edge effects for eight-deck games after one through four decks have been utilized.

Amount of edge reduction due to ace removal as shoes are depleted decks previously utilized (in discard rack)

   aces removed	  1	  2	  3	  4
	0	2.712%	3.166%	3.802%	4.757%
	2	2.543%	2.968%	3.564%	4.459%
	4	2.373%	2.770%	3.326%	4.162%
	6	2.204%	2.572%	3.089%	3.865%
	8	2.034%	2.374%	2.851%	3.567%
	10	1.865%	2.177%	2.614%	3.270%
	15	1.441%	1.682%	2.020%	2.527%
	20	1.017%	1.187%	1.426%	1.784%
	25	0.593%	0.692%	0.836%	1.040%

Compare these figures with the 2.373 percent nominal value for eight decks, remembering that the overall edge in the game for perfect basic strategy is about 0.5 percent. Ignoring the impact of aces on other factors, edge reduction owing to blackjacks over 2.373 + 0.500 or 2.873 percent gives players an advantage in the game. This would occur if fewer than four aces are gone when the shoe is two decks down, fewer than eight have been discarded when three decks are used-up, or fewer than 13 (not specifically shown) are gone when the shoe is four decks light.

The converse is that more aces drawn than would be expected from the law of averages gives the house additional advantage. Pretend you count 15 rather than eight aces expended when two decks are in the discard rack. Still assuming the original proportion of 10s, the table indicates that edge reduction due to blackjacks is 1.682 percent. This is 2.373 - 1.682 or 0.891 percent shy of the full-shoe value. Add 0.891 to the nominal 0.500 percent edge to find yourself a 1.391 percent underdog.

Does this suggest a card counting method less susceptible to distractions than the tried and true procedures? Is it as good? Do you play long enough for small edge shifts to make a positive difference in your results at the end of the day? Here's how that parabolic poet, Sumner A Ingmark, pontificated over such puzzles:

Accept what advantage you can get,
Aware that its impact on your bet,
May show in the long run, but not yet.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.