INSURE A GOOD BLACKJACK HAND? PART I

Should you insure a good blackjack hand? Blackjack gurus ridicule this question, replying that insurance is a side bet that has nothing to do with the player's hand. If more than one-third of the unseen cards are ten-valued then you insure; if less, you don't.

But what if the tens make up exactly one-third of the unseen cards? That makes the 2 to 1 insurance payoff exactly right, with no advantage to the casino or blackjack player. At first glance it seems that taking insurance in this case is wrong. It's like taking the odds in craps; you increase your bankroll fluctuations without any long run gain.

But wait. Let's look at the statement that the insurance bet has nothing to do with the original bet. This is not true, because correlation is involved. If you have a natural, the correlation is perfectly negative, -1.0. Whichever bet wins, the other loses. If you do not have a natural but the dealer does, then the negative correlation is also perfect: You lose the original bet and collect on the insurance. But what if neither you nor the dealer has a natural? Now the correlation between the lost insurance bet and the result of the original bet depends on the quality of your hand. If you have a 20, the correlation will be highly negative: The insurance bet is lost, and the original bet will probably win. With a 16, however, the correlation will be positive: The insurance bet loses, and the original bet will probably lose too.

These correlations lead to some interesting conclusions when there are exactly one-third tens in the deck. If you have a natural, then taking insurance should be automatic. It costs you nothing in the long run, and reduces bankroll fluctuation. If you have a 20, it seems to me that the decision should be the same. You will probably win the hand if you lose the insurance, so insuring to reduce fluctuation seems like a good idea. With a 16, however, bankroll fluctuation is increased, not decreased, by the fair insurance bet. I speculate that a player hand of 11, 19, or 20 should take the fair insurance bet, but other blackjack hands should not. Do any mathematicians out there care to comment? ♠