Why breaking up is hard to do... pairs, in blackjack, especially

Basic Strategy in blackjack is an exercise in optimization. It’s the set of decisions maximizing players’ statistically expected gains on favorable and minimizing expected setbacks on inauspicious hands. “Expected,” not in the sense of a specific amount anyone might gain or drop in a particular coup, but an average computed using probabilities with the laws of large numbers.

Once a person is familiar with casino blackjack, much of Basic Strategy coincides with intuition. Much, but not all. When dogma and instinct conflict, even experienced players aren’t always comfortable following “the book” rigorously and may stray from the straight and narrow.

The divergence sometimes develops when Basic Strategy says to take a card and gut feel to stay put. These tend to be situations where both choices have negative expectations – projected losses. Hitting lowers theoretical setbacks over many instances of the same hand. Standing avoids losses on the spot by busting. The dilemma is most common with totals of 15 or 16 against 7- through ace-up. Players realize the odds are they’ll lose to a bust if they draw and to a pat dealer’s hand if they don’t. Some bettors also balk at the rule about hitting 12s against 2-up or 3-up. They wonder whether to risk busting because they erroneously equate 2- or 3-up with 12 or 13 total and think the dealer is as weak as they are. Here are a few values of expected loss per dollar bet in 8-deck games for situations like these: 9-6 versus 9-up – $0.54 standing and $0.47 hitting, 7-5 versus 2-up – $0.29 standing and $0.25 hitting, and 10-2 versus 3-up – $0.25 standing and $0.23 hitting.

Some Basic Strategy doubles also instill doubt. These circumstances invariably have positive expectations for doubling as well as for their intuitive alternatives – usually hitting or standing. Either is good but one is better. Expectation for profit per dollar bet at the start of the round for the strongest sanctioned double, 6-5 versus 6-up, is $0.68 compared with $0.34 by hitting; for the weakest Basic Strategy double, A-6 versus 3-up, it’s $0.06 compared with $0.03 by hitting. The primary reason solid citizens occasionally forego doubles is anxiety about placing extra money at risk during the course of the round. A second rationale is the thought that on some hands, after receiving one card, they’d take another were it permitted. For instance, when doubling on 5-4 versus 6-up and finishing with 11, or on 6-4 versus eight-up and ending with 16.

Splits are the most frequently disobeyed Basic Strategy dicta. This, since players must raise what’s at risk during a round but may not perceive any benefit, or may even think their positions are worsening. The enigma arises in all three of the scenarios for which splitting is the rule.

A pair has negative expectation for standing or hitting but is theoretically profitable when split. The incentive to reverse a negative expectation should be compelling; however, the potential improvement may not be evident. A 6-6 against 5-up is an illustration. The pair is obviously weak when treated as a 12, but it’s not apparent that a six alone has an advantage facing 5-up. In fact, per dollar bet when a round begins, a 12 is forecast to lose an average of $0.16 and $0.19 by standing and hitting, respectively, while splitting comes in at $0.10 profit. All told, 20 hands meet this criterion: A-A/A, A-A/10, A-A/9, 2-2/7, 8-8/7, 3-3/6, 6-6/6, 7-7/6, 8-8/6, 2-2/5, 3-3/5, 6-6/5, 7-7/5, 8-8/5, 2-2/4, 3-3/4, 7-7/4, 8-8/4, 8-8/3, and 8-8/2.

A pair has positive expectation for standing or hitting, as warranted, but is predicted to win more when split. The total represented by the pair may seem strong, leaving players to wonder whether they should just take the money and run or if enough is to be gained by matching the amount already at risk and splitting the pair. An example is 9-9 versus 8-up. Standing, prospects are to earn an average of $0.10 per dollar bet at the start of the round. Splitting, the statistical profit is $0.23. In all, 16 hands are in this category: A-A/8, 9-9/8, A-A/7, A-A/6, 2-2/6, 4-4/6, 9-9/6, A-A/5, 4-4/5, 9-9/5, A-A/4, 9-9/4, A-A/3, 9-9/3, A-A/2, and 9-9/2.

A pair has negative expectation for standing, hitting, and splitting alike. Despite the extra dough up for grabs, though, average losses are least by splitting. These opportunities for splits are the most regularly ignored. The initial total is clearly bad, but intuition suggests that starting with the constituent cards may be twice as painful. The classic illustration is 8-8 versus 10-up. Splitting leaves players with an expected loss of $0.48 per dollar wagered at the start of the round. Next best, when allowed, is surrender with a certain loss of $0.50 on the dollar. Hitting and standing follow, each anticipating losses just under $0.54 on the dollar – the latter being fractionally worse. In total, 16 hands are in this class: 8-8/A, 8-8/10, 8-8/9, 9-9/9, 8-8/8, 3-3/7, 7-7/7, 6-6/4, 2-2/3, 3-3/3, 6-6/3, 7-7/3, 2-2/2, 3-3/2, 6-6/2, and 7-7/2.

The devil in all these situations is in the statistical details. Intuition goes just so far. And, although a player can win a round doing almost anything, over a period of time, the mathematics associated with the laws of probability triumphs. The poet, Sumner A Ingmark, said it like this:

‘Though intuition oft encumbers,
The truths are lurking in the numbers.