Assuming that the god of twenty-one is impressed by this sniveling appeal, he?ll give the wretched gambler a blackjack. But then the devil steps in to test our man?s new found faith ? he gives the dealer an ace as up card! TOB

EV maximizing index

For buying insurance, we have been told that any player who?s trying to maximize his/her expected value should take the bet if and only if:

{1} p>1/3 = 0.3333

As a corollary we get the insurance expectation as:

{2} mu = 3p -1

Setting mu = 0, we get the break-even point at p = 1/3, as expected.

We can illustrate this a little bit with the aid of an artificial remaining subset that has been altered to fit what supposedly is ?normal? composition, without forgetting the uncorrelated 7s, 8s and 9s (zero-valued cards), which we adjust accordingly, as well.

A normalized remaining subset

Table A

A 2 3 4 5 6 7 8 9 T

16 12 12 12 12 12 13 12 13 57

AT v A

After removing two aces and a ten from the above subset, we have:

p = 56/168 = 0.3333 exactly

Hi-Lo RC = 10, thus,

TC =10/ (168/52) = 3.0952

Let?s compute the expected value for our BJ hand without buying on insurance.

Computer output?

Win = 0.6667
Ties = 0.3333

An elementary 0.6667*1.5 = 1

EV = 1

Exactly the same as taking Even Money; thus we accept the offer while lowering the variance at the same time. Well in accordance with Griffin?s advice:

When confronted with two courses of action with identical expectations, prefer that one which reduces the variance.

Exact calculations for any given subset tend to show very slight (see Cacarulo?s ?Insurance Charts for Hi-Lo and K-O?) discrepancies for this break-even figure as a function of different depths where the calculations are performed. The six-deck variety is a good example where these minuscule differences show up.

It is interesting to note that I have seen at least five ?exact? critical borders, among different expert opinions for the Hi-Lo point count. Employment of combinatorial analyzers, with the aid of EOR?s and algebraically derived formulas, certain assumptions, and straightforward math calculations. You name it!

Partial insurance for the Kelly bettor.

TOB: a Kelly bettor should consider insuring at least a portion of his blackjack against a dealer ace if :

{3} p>1/3(1 + f)

Thus players concerned with optimizing their average logarithm of wealth are already prepared to buy-in on negative-expectation insurance bets!

f = Kb% Ins. for less if p>

Quarter 0.0025 0.3325

Half 0.0050 0.3317

3 quarters 0.0075 0.3309

Full Kelly 0.0100 0.3300
0.0200 0.3268
0.0300 0.3236
0.0400 0.3205

Let?s assume a 6 deck with das and 75% dealt out, maximizing at TC = 4, and using optimal bets. Figures like one and two percent of the total bankroll

tend to make a lot of sense here for a Kelly bettor ?flirting? with bets round the TC = 3 level.

Griffin: The correct proportion of the blackjack to insure for these Kelly bettors is :

{4} x = 3p + (3p-1)/f

Note that {2} is the insurance wager expectation and thus Griffin?s derivation is rational.

Now we have our five players holding BJs versus the dealer?s ace, and all of them below the critical point for insurance, performing speed-of-light calculations in order to get their correct partial insurance amount to be wagered. Right? Wait a minute. Read on.

Insure a BJ for All or Nothing

No problem. Let?s see what we get then:

TOB :If it?s a choice between insuring all or nothing, insurance should be taken if :

{5} p > 1 ? (log (1 + f))/ (log (1 + 3/2*f))

and so we get:

Kb fraction tens density

0.0025 0.3329
0.0050 0.3325
0.0075 0.3321
0.0100 0.3317
0.0200 0.3301
0.0300 0.3285
0.0400 0.3269

As you can see inside DD?s archives, the RA figure for taking insurance for this particular hand, which BJ expert Cacarulo has given us, is 2.82,

well in line with the tens density we have here, applying Griffin?s formulae. You should focus the attention on the 1% fraction, mainly.

Eg., if we remove an extra ten from the above artificial remaining-cards subset (Table A), we get:

p = 55/167 = .3293

tc = 9/ (167/52) = 2.8024

Kudos for him. Our results here should be viewed as a sort of double-check for the RA index.


Fancy memory! Where did I read long ago something quite similar to:
We tend to fall in love with the theory more often than not, so as to lose sight of its practical perspectives.

But the guy who wrote that surely speaks the truth. So, in order to avoid falling in the trap here, I feel obliged to share a bit of final advice:

Forget the partial insurance bets on your BJ. These bets are either difficult to evaluate properly and /or not the usual gambler?s behavior, and thus are subject to suspicion. Take even money always on your BJ anytime you have a TC of 2.8 or higher with the Hi-Lo count. Try to capitalize on these almost break-even bets by calling them to the attention of any pit-critter camping around. Just seek his/her advise. Bet you life savings that he/she will say: ?Of course, you should.? By doing this, you?re enhancing both your longevity and the general logarithm of your wealth. Two birds, just one stone!




Readers with math abilities or the inclination to make some further research on this interesting topic of break-even, or almost-even insurance bets expectations should be cautious with comparative tests based on Certainty Equivalent. Slight modifications to the published formulas in BJA3 are on the way. You can?t simple copy the ones form page 371. Ditto for the critical fraction from page 372. The reason is the strong correlation between the hand and the insurance bet.

When dependence is assumed you have:

Var (x + y) = Var (x) + Var (y) + 2Cov(x, y) which is equivalent to

Var (Hand + Insurance) = Var (h) + Var (i) + 2*Cov (h, i)

Obviously, with dependence, we have an elementary addition for the expected values:

EV (x + y) = EV(x) + EV(y) therefore

EV (h+i) = EV (h) + EV (i)

So our formula for CE seems to be:

{6} CE(h + i) ={ev(h) +ev(i)}*f ? {(Var(h) + Var(i) + 2*Cov(h,i)}*f/2

That needs to be compared for the CE case when we decline the offer for the insurance bet and we?re evaluating the Certainty Equivalent of the hand only. The one with the higher CE should be selected for players trying to maximize their utility bets. Hard task to use such formulas, but surely not rocket-science. It can be done.

Insure a BJ seems to be the easiest for covariance calculations, unfortunately necessary when choosing the above formula.

Corr = Cov (h, i)/ (std (h)*std (i))

With total correlation (rho = 1), we simply have

Cov (h, i) = sqr (Var (h))* sqr (Var (i))