If You Can't Predict the Future, Here's What's Next Best

At my blackjack table recently, a player drew a four-five versus a dealer seven-up. "Double or hit?" the player asked. "Double down," the dealer answered. The rest of us disagreed, arguing that basic strategy says to hit under these circumstances.

The player hit, pulling a 10 to make 19. The dealer turned an ace for a total of 18. Doubling would have paid twice as much with identical cards. "I should'a doubled," the solid citizen sulked.

Basic strategy gave the right decision, although doubling would have paid more in this instance. The story has three morals.

Moral 1: Dealers don't necessarily know how best to play. Skill on one side of a table may not mean aptitude on the other. Nor do eons of observation help casino personnel predict random events.

Moral 2: The quality of a gambling decision is based on what's known in advance, not afterward. It would be correct to hit 18 against eight knowing that a three was next or the hole card was an ace. But, seeing the 18 and the eight, and knowing only the probabilities of what's next and in the hole, the best decision is to stand - even if both the three and the ace later turn up.

Moral 3: Versions of basic strategy for particular blackjack rules give the optimum decisions for any combinations of player hands and dealer up-cards. I'll illustrate using the nine versus seven-up situation. Basic strategy says to hit. Why is this better than doubling?

Hitting - continuing to draw until the hand reaches or exceeds 17 - yields chances of 53.5 percent to win, 36.2 percent to lose, and 10.3 percent to push. This is a 17.3 percent player advantage, a net expected win of $1.73 on a $10 bet.

Doubling - making an additional bet but drawing only one card - yields chances of 49.1 percent to win, 45.8 percent to lose, and 5.1 percent to push. This is a 3.3 percent player advantage, a net expected win of $0.66 on a $10 bet doubled to $20.

Expectation is positive either way. But the edge is so much greater when hitting than doubling in this case that it overshadows the benefit of the extra bet in a favorable situation.

Also to be considered are the bankroll swings characterizing each alternative. Bet $10 on nine versus a seven-up. Fluctuation is $9.34 for hitting and $19.47 for doubling. These values swamp the corresponding $1.73 and $0.66 expected wins during typical sessions. But, is the much greater fluctuation associated with doubling enough to shift the relative advantage of the bets?

For an answer, consider normal bankroll ranges after 100 decisions. Hitting nine versus seven-up, a player has over 50 percent chance of being from $40 to $310 ahead and more than 75 percent chance of being from $10 to $360 ahead. Doubling yields over 50 percent chance of being from $210 behind to $340 ahead and more than 75 percent chance of being from $330 behind to $460 ahead. Doubling therefore raises the potential gain in this situation. But it also heightens the risk of winning less - or even losing.

A further point is whether short-term swings are more apt to be up or down with either tactic. This can be addressed by examining skewness. Hitting has a skew of ?0.332, moderately favoring upswings. Doubling has a skew of ?0.066, barely favoring upswings.

What's best? The edge gives hitting nine versus dealer seven a greater expected win than doubling. Fluctuation yields higher potential upswings with doubling than hitting, but at the risk of correspondingly lower downswings. And skew shows a stronger tendency toward an upswing for hitting than doubling. Hitting is therefore more promising by two measures, and less risky by the third, even though doubling wins more often than it loses.

Sumner A Ingmark, a poet who always knows how his epics will end because he writes the last verses first, said it this way:

It takes no skill to play with a sure hand,
You need finesse when you've drawn a poor hand,
'Cause you don't know the future beforehand.