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# Are There Conditions Under Which Insurance Is a Good Bet?

28 June 2004

Most blackjack players have heard "you're not supposed to take insurance." Many do, anyway. This auxiliary bet, equal to half your original wager, is offered when the dealer shows an ace. The payoff is 2-to-1 if the dealer has a 10 in the hole. Therefore, a win nets you even money on blackjacks of your own, and pushes other hands. A solid citizen may justify insurance emotionally, especially when a sure return on a blackjack seems attractive. But be aware that under most conditions, statistically, it's the costliest of the common departures from Basic Strategy.

Consider the case when you ignore any information at your disposal except shoe size and the fact that the dealer has an ace-up. An eight-deck game would have 8 x 16 or 128 10s out of 415 unseen cards (8 x 52 or 416 cards minus the dealer's ace). You win \$2 for every \$1 bet, so the edge is equivalent to \$2 x (128/415) - \$1 x (415-128)/415. The arithmetic yields is a loss of 7.7 cents on the dollar. That's 7.7 percent for the bosses.

Overall, the impact of insurance on house advantage depends on how often dealers are expected to show aces. With six and eight decks, insuring every instance of a dealer's ace imposes penalties of 0.28 and 0.29 percent, respectively. Insuring only your own blackjacks and two-card 20s costs 0.014 percent and 0.034 percent, respectively. These handicaps may seem small. But the edge for perfect Basic Strategy is a mere 0.4 to 0.5 percent depending on the rules, so the relative increase is substantial.

As weak as insurance is for a Basic Strategy player, it's one of the strongest weapons in a card counter's arsenal. If, having tracked discards during a shoe, an individual knows that 10s comprise over a third of what's left, insurance becomes a favorable proposition. The "normal" proportions of cards are 30.8 percent 10s and 69.2 percent other. At 35-65, a card counter expects earnings of \$2 x 0.35 - \$1 x 0.65 or \$0.05 per dollar; 40-60 brings this to \$2 x 0.40 - \$1 x 0.60 or \$0.20 per dollar.

In principle, this effect can be exploited without counting cards, based on what's exposed in a face-up game when the round is dealt. A dearth of 10s on the table mean more still in the shoe. Further, the more spots in action, the greater the depletion of the shoe and the fewer total cards remaining. Together, these factors raise the proportion of 10s to be drawn.

In practice, with four or more decks and seven or fewer spots, the effect isn't enough to overcome the normal house advantage. At eight decks, for instance, if seven spots are dealt with no 10s, the casino retains the edge although it drops to 4.2 percent. With six decks in the same situation, it's 3.0 percent. Profitable prospects are possible at three decks and below. The accompanying table shows edge for insurance in 1-, 2-, and 3?deck games with one to seven spots in action when no 10s are exposed.

Edge on insurance with no 10s on the table
(minus favors the house, plus favors the player)

 active spots onedeck twodeck threedeck 1 -2.0% -5.0% -5.9% 2 +2.1% -3.0% -4.6% 3 +6.7% -1.0% -3.4% 4 +11.6% +1.1% -2.0% 5 +17.1% +3.2% -0.7% 6 +23.1% +5.5% +0.7% 7 +29.7% +7.9% +2.1%

The values in the table look promising. However, although "continuous shuffling machines" typically use three decks, face-up games of one and two decks are rare. Further, the chance is small of encountering a round in which the dealer has an ace and the required numbers of hands are dealt without a 10 appearing. In a three-deck game, the probability of six players with no 10s, giving you 0.7 percent edge, is low at about one out of 1,360. Assuming you could find a one-deck game dealt face up, the huge 29.7 percent advantage with seven spots in action can be expected to occur on the average of once every 7,244 rounds.

So, watch for the opportunity, but don't accuse the casino of a conspiracy if it doesn't come along regularly. It's as the counters' Kipling, Sumner A Ingmark, observed:

Don't pin your hopes on circumstances rare,
But be alert to find them when they're there.

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Best of Alan Krigman
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.