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Best of Alan Krigman

Gaming Guru

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Should blackjack players say "yes" to insurance when the gurus say "no"?

27 December 2010

Anybody who indulges in casino blackjack is, or should be, aware of the cardinal rule never to take insurance. A reasonable fraction of bettors flout this fiat when they have blackjacks themselves; they reason in qualitative "utility" parlance that they prefer the certainty of even money to a gamble between a 1.5-to-1 payoff and a push. Somewhat fewer players take insurance when they have 20s, figuring their hands are strong enough to warrant trading a push rather than a loss if the dealer has a blackjack for a good shot at a half-bet win otherwise. And a relatively small cadre pony-up when they see a dealer's ace-up regardless of their hands.

Insurance isn't a free ride. Full understanding of the price requires distinguishing between absolute and conditional probabilities and edges. Absolute values are those faced before a round is dealt, and involve the chances of players receiving particular hands and dealers simultaneously having ace-up and 10 in the hole. Conditional values are those faced after the deal when players know their hands, see the ace, and decide whether to bet that the dealer will flip over a 10.

The absolute case gives the impact of insurance on the overall edge in the game. The figures are most helpful to frequent players concerned with their prospects over extended action. Here, the penalty for always insuring your own blackjack is 0.014 percent; this may seem cheap enough to guarantee an even-money win, but it's a 3.14 percent increase over the edge for perfect Basic Strategy. The reduction in edge for consistently insuring 10-10 is 0.031 percent – a 7.05 percent net rise over the edge otherwise attainable. On ace-9, the damages amount to 0.003 percent, adding 0.713 percent to the nominal edge. Routinely insuring every hand drives up edge by 0.287 percent, a 65.67 percent increase relative to what optimum play gets you.

Most individuals are more acutely attuned to the present than to their blackjack lifetimes. The quandary is then, given such and such a hand against a dealer's ace, what's the chance of winning an insurance bet and how much do the bosses charge for it in edge? The answer depends on what's left in the shoe. Assume you don't count cards or bother to check what's on the table in front of the other players. The only things you know are the shoe size and the fact that three cards have been withdrawn – your two and the dealer's ace.

Make believe you're in a six-deck game with a blackjack when the dealer has ace-up. There are (6 x 52) - 3 or 309 unknowns. The 10 in your hand is gone from the six decks so (6 x 16) - 1 or 95 are left. This means the chance of a 10 in the hole is 95/309 or 30.74 percent. Since the payoff is 2-to-1, house edge on insurance at this juncture is (0.3074 x 2) - (0.6926 x 1) or 7.77 percent.

Pretend, instead, you have 10-10 versus ace-up. Now there are (6 x 16) - 2 or 94 10s in the 309 unknowns. The chance of the dealer having one of them in the hole is 94/309 or 30.42 percent. And the bosses have an edge of (0.3042 x 2) - (0.6958 x 1) or 8.74 percent.

For any other two-card combination which includes a single 10, the calculation is the same as for a blackjack; chance of winning is 30.74 percent and edge is 7.77 percent. When your hand contains no 10s, 96 cards of this value are theoretically available to be drawn from 309 unknowns, leading to a probability of 96/309 or 31.07 percent and an edge of 6.70 percent.

From the conditional perspective, insurance looks worse than it does in terms of the absolute or overall impact on the game. But, more, knowing the conditional probabilities, you can ask yourself the utility question quantitatively rather than qualitatively. For instance, suppose you have a blackjack against an ace-up. You don't have to speculate vaguely whether "the certainty of even money is preferable to an unspecified gamble between a 1.5-to-1 payoff and a push." You can conjecture precisely whether "the certainty of even money is preferable to a 60.26 percent chance of a 1.5-to-1 payoff versus a 30.74 percent chance of a push." Or, alternately, whether you consider "paying an edge of 7.77 cents per dollar to be guaranteed even money better than a 60.26 percent chance of a 1.5-to-1 payoff versus a 30.74 percent chance of a push."

Of course, there may be extenuating circumstances that override the numbers at any point of a session. Say you got a blackjack on a huge wager for which you'd be thrilled to get paid 1-to-1, or on a bet that will finally boost you over the top after a monumental struggle if it gets you even money but will still leave you behind on a push. Certainty of even money may then be worth more to you than a 1.5-to-1 versus push tradeoff, no matter what the gamble. You'll recall that Aesop didn't mention probability in the moral of the fable of the Hawk and the Nightingale, that "a bird in the hand is worth two in the bush." Or, as the beloved bard, Sumner A Ingmark, wrote:
A potential reward may possess great allure,
When compared with a prize that is lesser but sure,
Since the gamble may cause you to walk away poor.

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.