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# When Insurance Gives Blackjack Players an Edge

27 October 1997

Players can "take insurance" at blackjack when the dealer shows an ace. This bet can be anything up to half the original wager. Insurance pays 2-to-1 when the hole card is a 10 and the dealer has blackjack; it loses otherwise. Solid citizens who stick strictly to basic strategy know insurance is a "sucker bet." Every book, wallet-size crib card, and gaming guru says it is.

Except, sometimes it isn't. I'm not alluding to the endless arguments about insuring your own blackjack, which pit mathematical probability against psychological utility. I mean circumstances when insurance inherently favors the player.

Picture an eight-deck game. The shoe contains 52x8=416 cards. Of these, 4x4x8=128 are 10-valued and the remaining 416-128=288 are ace through nine.

What are the odds against the dealer having a 10 in the hole... a blackjack? They're 288-to-128 or 2.25-to-1. But a winning insurance bet only pays 2-to-1. The disparity between 2.25-to-1 and 2?to-1 gives the house an edge or advantage on the bet. Think of the edge as a fee the casino charges to book a wager. On insurance, the numbers work out to 7.7 cents per dollar bet.

For taking insurance to be favorable, the odds against winning would have to be less than 2-to-1. Surprise! Basic strategy blackjack buffs can find situations when this occurs.

As an illustration, say you're in a multi-deck game and it's late in the shoe. Before the round is dealt, you eyeball the discard rack and estimate five decks out, three left. Six spots are in action. When the cards are dealt, you see thirteen of them - two for each bettor, one for the dealer. The dealer has ace-up and offers insurance. No 10-values appear on the layout.

Now, what are the odds? The three decks in the shoe before the deal contained 52x3=156 cards. Without knowing what was dealt earlier, you can figure this included 4x4x3=48 10-values. After the deal, 156-13=143 cards are still unseen. Since no 10-values were exposed, 48 remain along with 143-48=95 non-10s. The odds against the dealer having a 10 in the hole under these conditions are 95?to-48 or 1.98-to-1. Insurance, paying 2-to-1, is slightly favorable. The player has an edge of 0.7 cents per dollar bet.

If the shuffle point is deep enough so a round begins with only two decks - 52x2=104 cards - sitting in the shoe, insurance can be better yet. Assume, in such an instance, that cards are dealt to six players. The dealer draws an ace-up; no player gets a 10-value. You've seen 13 cards so 104-13=91 are unknown.

With no knowledge of previous play, the unknowns can be considered to include 4x4x2=32 10-values and 91-32=59 non-10s. The odds against the dealer having a blackjack are now 59-to-32 or 1.84-to-1. Since this is less than 2-to-1, it's a favorable proposition. The player has an edge of almost 5.5 cents per dollar bet.

Again with two decks remaining, insurance is favorable even if one 10-value appears in the cards dealt to six players. The unknown set of 91 cards then includes 31 10-values and 60 others. The odds are therefore 60-to-31 or 1.94-to-1, still under 2-to-1. Players have an edge of nearly 2.2 cents per dollar bet.

Some warnings are in order. (1) Positive expectation is a far cry from a guaranteed win. (2) If you underestimate how many decks remain in the shoe before the deal, you're apt to think you're favored when you're not. (3) Conditions for having an edge on insurance are relatively uncommon: shoe penetration often isn't deep enough for deals to be started with fewer than three decks remaining, rounds of initial hands rarely have one or no 10-values, five spots in action give you an edge only with two and a half or fewer decks before the deal and no 10s on the layout, and four spots requires two or fewer decks and no 10-values. Still, as the odester Sumner A Ingmark optimistically observed:

Two maxims winners oft obey,
Are "while the sun shines, make your hay,"
And "carpe diem - seize the day."

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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.