Insure a Good Hand?

Blackjack Insurance on Good Hands May Be A Good Idea After All

By Peter A. Griffin

(From Blackjack Forum Volume VIII #2, December 1988)
© Blackjack Forum 1988

[Note from Arnold Snyder: In the December issue of Blackjack Forum (Vol. VII #4), Marvin L. Master conjectured that if your card counting system indicated that the insurance bet was dead even, it may be advisable to insure a “good” hand, since this play would tend to reduce fluctuation. Marvin’s logic is clear. If the dealer does have a blackjack, then you will lose a bet you expected to win. Taking insurance would save this bet on one third of these hands, and on those hands where the insurance bet loses, you still expect to win your initial “good” hand. Thus, bankroll fluctuations are reduced. Here now, to lay this controversy to rest, is Peter Griffin’s final word on whether and when you should take insurance on “good” blackjack hands. More probably, this article will give nightmares to players who consider attempting to work out Griffin’s insurance formula when playing. Griffin shows that it is sometimes advisable to insure good hands—in order to reduce fluctuations—even when the insurance bet has a negative expectation! Unfortunately, most dealers only allow a couple of seconds for the insurance decision. So, the simplest answer is: Marvin was right! Insure your good hands when it’s a dead even bet.]

Marvin L. Master asks the question: Should you, to reduce fluctuations, insure a good hand when precisely one third of the unplayed cards are tens?

The answer depends upon what criterion for “reducing fluctuations” has been adopted. Griffin, in his monumental epic The Theory of Blackjack, shows that there are occasions when a Kelly proportional bettor would insure a natural with less than one third of the unplayed cards being tens.

Theoretically, this criterion could also be used to analyze whether to insure 20 and other favorable holdings. However, the answer is dependent upon both the fraction of capital bet and the distribution of the non-tens remaining in the deck.

An approximate calculation based upon what would seem a reasonable assumption in this regard suggested that 20 should be insured, but 19 not. Precise probabilities for the dealer were not computed, and the answer could well change if they were, or if a different fraction than assumed were wagered.

Another, more tractable, principle to reduce fluctuations also appears in The Theory of Blackjack: When confronted with two courses of action with identical expectations (the insurance bet here is hypothesized to neither increase nor decrease expectation), prefer that one which reduces the variance, hence average square, of the result.

This proves particularly easy to apply here. Let W, L and T stand for the probabilities of winning, losing, and tying the hand assuming insurance is not taken. In this case the average squared result is

ENx2 = 1 – T

If insurance is taken the average square becomes

EIx2 = 1/3 02 + W(1/2)2 + T(-1/2)2 + (L-1/3)(-3/2)2 = (W + T + 9L – 3)/4

Insurance will have a smaller average square if

W + T + 9L – 3 < 4 – 4T

Equivalently

W + 5T + 9L < 7

Or, subtracting

5(W + T + L) = 5

4L – 4W < 2

L – W < .5

L < W + .5

This will clearly be the case for player totals of 20, 19, 18, 11, 10, 9 and 8 if the dealer stands on soft 17. If the dealer hits soft 17, 18 would probably still be insurable, but not 8.

Returning to the Kelly criterion, the interested reader would be well advised to consult Joel Friedman’s “Risk-Averse” card counting and basic strategy modifications. Among Joel’s astute observations is that if a player confronts an absolute pick ’em hit-stand decision he should hit rather than stand. The reason is that he thereby trades an equal number of wins, (+1)2, and losses, (-1)2, for pushes, (0)2, thus reducing fluctuation. ♠