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Should you take insurance in blackjack without counting cards?

1 February 2010

A, if not the, prime directive of Basic Strategy in blackjack is never to take insurance. The reason is that, under normal circumstances, this bet gives the house a huge advantage.

Pretend, for example, you're in a six-deck game, know nothing about the proportions of ranks previously withdrawn from the shoe, and ignore what's in sight on the table at the moment. Here's how to figure the edge.

Since you haven't tracked what's gone before, cards used or to be played are equivalent un-knowns. From the six times 52 or 312 total cards, subtract one for the dealer's ace. Of the other 311, four ranks times four suits times six decks is 96 10-values, any of which could be the hole card. So the probability the dealer has a blackjack is 96 out of 311 – 96/311 or 30.868 percent; the chance of anything else is the complementary 69.132 percent.

Edge is payoff per dollar bet times the chance of winning minus the chance of losing. This works out to 2 x 0.30868 - 0.69132 or -7.395 percent (the minus sign indicates the house is favored). This is about 15 times the 0.5 percent overall edge in the game for a solid citizen following Basic Strategy.

Some players with blackjacks view insurance as a guarantee of even-money, edge notwithstand-ing. Well, de gustibus non est disputandum. But if you're in the game for the long haul and are concerned about edge, figure that your blackjack and the dealer's upcard mean two aces and a 10 are gone leaving 309 cards with 95 10-values. The chance the dealer has a blackjack is 95 out of 309 or 30.744 percent. Edge increases to 7.767 percent for the house.

You can get additional information about edge without formally counting cards, however. Check the cards dealt at the start of a round before a dealer with an ace-up offers insurance. The fewer 10-values in evidence, the higher the proportion left to be exposed and the more likely the dealer will have one in the hole.

The most promising case would be a table with six active spots and no 10s showing. In all, 13 cards are then visible – two on each spot plus the dealer's ace-up. This leaves 312 - 13 or 299 unknowns either in the shoe or the discard pile, of which 96 are 10-values. The chance of the dealer having a 10 in the hole is 96 out of 299 or 32.107 percent. House edge corresponding to this probability drops to 3.679 percent, but is still steep compared to 0.5 percent for the Basic Strategy game as a whole.

With eight decks, the no-knowledge probability the dealer will match an ace-up with a 10 in the hole is 128 (four 10-ranks times four suits times eight decks) divided by 415 (eight decks times 52 minus one cards) or 30.843 percent. The house edge works out as 7.470 percent, more than in six-deck games. Using what you see on the table with six spots in action, house edge when insurance is most apt to pay – 13 non-10s displayed – is 4.715 percent.

A one-deck game has the least house advantage on insurance. Out of 51 unknown cards, 16 are 10-values so the probability of a dealer blackjack with an ace-up is 16 out of 51 or 31.373 percent, and house edge is 5.882 percent edge. With one-deck, cards are dealt face-down, so you know what's in your own hand but nobody else's. If you're on one spot and have no 10s, the probability the dealer has a blackjack is 16 out of 49 or 32.653 percent and the house edge is 2.041 percent. If you're on two spots and have no 10s, the dealer's chance is 16 out of the 47 unknowns. This is 34.043 percent, which gives you a 2.128 percent edge and makes insurance an advantage bet. If you're on two spots and have one ten, edge reverts to the house and is 4.255 percent.

You might wonder what's the chance you'll have no 10s in two hands of a one-deck game when the dealer has an ace-up. It's 29.356 percent, so the opportunity, while appealing, arises but not too often. Unless, of course, your planets are properly aligned, such that you feel confident in flouting the laws of probability. But, as the beloved bard, Sumner A Ingmark, wrote:

Some folks spend their lives facing chance and disdaining it,
When asked what's the secret, they're baffled explaining it,
And more than they'd like, they have trouble maintaining it.

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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.