A Blackjack Bonus May Be No Bonanza

Casinos sometimes offer bonuses for certain blackjack hands. I don't mean giant jackpots awarded for extra money wagered on side bets. Rather, premiums such as automatic 2-to-1 payoffs for 7-7-7 or non-breaking six-card hands -- even if they push or lose.

Naturally, if you earn a bonus because of a lucky draw, you won't turn it down. So, offered the choice of tables with otherwise identical rules, go for the game with the bells and whistles.

But the situation isn't always this simple.

For one thing, rules might not be otherwise identical. The choice may be six decks with no bonuses but resplitting of pairs, or eight decks paying 3-to-1 for a red 6-7-8 where pairs can be split only once. Calculating the optimum options may demand the mathematical mind of a Nikolai Ivanovich Lobachevskii. Moreover, one answer may satisfy orthodox bettors, another may please solid citizens who trust hunches more than the laws of probability.

In general, however, fewer decks in play, ability to resplit pairs, less constraint on doubling, and surrender -- individually and certainly in combination -- are more valuable than most bonuses. The main exceptions are 2-to-1 payoffs on uncontested blackjacks. The benefit here is greatest if all naturals get the prize; it diminishes rapidly if the bonus is paid only on same-color, same-suit, designated-suit, or specific blackjacks.

A second complicating factor is that bonus situations don't arise as often as you may imagine. Consider a suit-insensitive 6-7-8. The chance of an appropriate start -- 6-7, 6-8, or 7-8 -- is only 3.55 percent. Hitting such a hand trying for the bonus has 7.69 percent probability of success. Overall, the likelihood of three cards forming the winning combination is a scant 0.27 percent.

The better blackjack bonuses have much greater frequencies of incidence. The chance of any blackjack is roughly 4.7 percent. A single-suit blackjack -- two spades, hearts, clubs, or diamonds -- has about 1.2 percent chance of occurring. For any particular blackjack, such as ace and jack of spades, probability falls to a scant 0.3 percent -- three chances out of a thousand. Of course, you'll rarely see anyone offering the 4.7 and 1.2 percenters.

Another problem emerges when a hand which normally calls for standing is hit in trying for the dividend. I'll illustrate with a bonus for 6-7-8 regardless of suit.

Say you start with 7-8 versus 8-up. You're swimming upstream, but hitting always outshines standing and the possible bonus reduces the penalty. Here are the figures. With this hand, you're ordinarily anticipating a loss of about $0.41 per dollar bet. A 2-to-1 bonus drops expected loss to $0.33 per dollar. A 3-to-1 premium cuts this down to $0.26 per dollar bet.

What if, instead, you have a 7-8 versus 2-up? With no bonus, the hand is an underdog either way. Standing is the lesser of the evils. It gives an expected loss of about $0.29 per dollar bet, while hitting jumps the penalty to roughly $0.42. With a 2-to-1 bonus, the expected loss for standing stays at $0.29. That for hitting drops to $0.35 per dollar -- still more burdensome than standing, so going for the bonus increases the house edge in the game. At 3-to-1, hitting in this situation would beat standing by a fraction of a cent per dollar and becomes the preferred play.

The dilemma deepens if the dealer starts with 6-up. Expected loss standing on 7-8 is $0.15 per dollar. Hitting with no bonus boosts this to $0.37. At 2-to-1 for 6-7-8, the penalty decreases to $0.28. Even at 3-to-1, it's $0.20, again the worse move.

An old Bezonian proverb says the sizzle sells the steak. The poet, Sumner A Ingmark, demurred -- at least for casinos outside old Bezonia -- thusly urging bettors to quest for true quality: