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# Why Many Seasoned Players Insure their Blackjacks

10 February 2004

The house gets a high edge on insurance bets in blackjack. So most gaming gurus tell players not to make them. But lots of solid citizens, including many seasoned veterans, do so anyway. Especially when they have blackjacks themselves and insurance is a guarantee of "even money." Why is the rule so often flouted?

Think about this hypothetical situation. A rich uncle hands you \$50. Then he asks if you'd like to keep it, or bet it on some known chance of winning \$75 versus the complementary probability of giving it back. When the likelihood of winning is sufficiently high, say 95 percent, the majority of people would go for the gamble. When it's low enough, for instance, 40 percent, most folks would pocket the \$50 under ordinary circumstances. At what probability level would you draw the line for yourself? Would a gamble for \$75 be more attractive than a sure \$50 if the chance of success were 80 percent? How about 70 or 60 percent?

Blackjack buffs face just this dilemma with a \$50 bet when they get a natural and the dealer shows an ace-up. They can take insurance and receive a certain \$50 profit. If they decline, they have slightly under 70 percent chance of winning \$75 and somewhat over 30 percent chance of neither gain nor loss. Expected value offers an alternate view of the situation. Insurance gets players their \$50 back and a \$50 profit for \$100 total. With the gamble, the expected value of the bet works out to \$101.92. It's a choice between an assured \$100 or a gamble pegged at \$101.92.

Probability theory coupled with the goal of maximizing expectation tells you to avoid insurance. This is Basic Strategy. Utility theory, which accounts for personal preferences, explains why this bet is so popular. It asks, albeit implicitly, whether 70 percent is a strong enough or still too weak a prospect to gamble for \$75 rather than accept the certain \$50, how appealing \$75 is relative to \$50, and -- at a higher level of sophistication -- if the extra \$1.92 justifies the risk. The certainty often comes out ahead in the answers.

Insurance is more enigmatic when you have other than a blackjack and the dealer has ace-up. The probabilities of various dealer downcards are still somewhat more than 30 percent of a 10 for a blackjack and a bit shy of 70 percent for anything else. However, the choice is of a gamble either way. Further, the range of outcomes is wider and comparisons tougher to conceptualize.

Taking insurance, the chance of a push is 30 percent since your bets will wash out if the dealer has a blackjack. But, the remaining 70 percent of cases can result in a half-unit gain if you win, a half-unit loss if the hands push, and a 1.5-unit loss if the dealer wins. The probabilities of the various results are shown in the accompanying table for starting hands of 17 to 20.

Chances of resolving player starting hands
when the dealer has a "playable" ace-up

 starting hand win push lose 17 11.5% 13.1% 44.6% 18 24.6% 13.1% 31.5% 19 37.7% 13.1% 18.4% 20 50.7% 13.1% 5.4%

Without insurance, you pick up one unit on a win and are neutral on a push, with probabilities as given in the table. You drop one unit if the dealer wins, regardless of how, the chance being the playable loss shown in the table plus 30 percent for a blackjack.

These factors make utility considerations more complex when you don't have a blackjack. For example, assume you bet \$50 bet, pull a 19, and the dealer shows an ace. Rounding off values, insurance gives you 30 percent chance of breaking even, 38 percent of winning \$25, 13 percent of losing \$25, and 18 percent of losing \$75. With no insurance, the figures are 38 percent of winning \$50, 13 percent of pushing, and 48 percent of losing \$50.

Utility is easy to invoke on the insurance decision when you have a blackjack. When you don't, the data in the table can still help since you know the amounts involved and can see the chances. Much like the poet, Sumner A Ingmark, pictured when he penned: