Should You Take Insurance When You Have a Blackjack?

Here's a puzzle that's perplexed blackjack players since time immemorial, maybe longer. You get an ace-queen - a fat and sassy blackjack. The dealer gets an ominous ace-up and says, "insurance is open." Do you take insurance... or your chances?

Ask a battalion of bettors and you're apt to receive all the following answers. 1) I take insurance whenever I have blackjack; that way I can't lose. 2) I never take insurance; it's a sucker bet. 3) I always take insurance. 4) I take insurance on a "true count" of +2 or higher. 5) I take insurance when hardly any 10s are showing; they're due, so the dealer's bound to turn one over. 6) I take insurance when lots of 10s are showing; they're coming out, so the dealer's bound to have one in the hole. 7) I insure strong starting hands; it's a good hedge. 8) I insure weak starting hands; it's my best shot at not losing the bet. 9) I take insurance when the dealer is running hot. 10) I take insurance when I have a hunch the dealer's gonna pull a blackjack.

Answers 5) through 10) are bettors' balderdash while 3) and 4) raise issues beyond simply insuring blackjacks. So, for now, I'll focus on points 1) and 2) - assuming standard six- or eight-deck shoes with cards in approximately pristine proportions.

What do the laws of probability say to expect? Averaged over many hands, for every 13 dealer aces-up, four will yield blackjacks and nine will not. That is, the probabilities are 31 percent of the dealer having blackjack and 69 percent something else.

Say you bet $10. You get blackjack. The dealer shows ace-up. Take insurance and you're assured of a $10 win. Decline it and your chances are 31 percent of a "push" and 69 percent of a $15 win.

Translate this into dollars by assuming 13 instances of the situation. With insurance, you're locked into $10 all 13 times, a total of $130. Without insurance, you could win up to $15 all 13 times, a total of $195. Or you could tie 13 times and win nothing. Both extremes are unlikely. Instead, your statistical expectation is to win $15 in nine cases and tie in four, a total of $135. Expected win, a factor based purely on house edge, theoretically penalizes you $5 - half a bet - for every 13 times you insure a blackjack. That's $0.038 per dollar wagered.

But, casino gambling goes beyond expectation due to edge. It also involves volatility, the characteristic swings that boost solid citizens over the top or knock the props out from under them.

A sure $10 per hand, $130 in 13 hands, has no volatility; fluctuation relative to the expected win is zero. The fluctuation associated with the chances of winning nothing or $15 on an uninsured blackjack is $6.92 per bet; the fluctuation associated with the aggregate expected win of $135 on 13 bets is $24.95.

What does this mean in terms of bankroll? According to an obscure but useful rule of mathematical statistics, the "Chebychev Inequality" (honest, I'm not making this up), you have roughly:
o 75 percent chance of winning under $105 or over $165,
o 50 percent chance of winning under $100 or over $170,
o 25 percent chance of winning under $85 or over $185.

One more question. Are the up- and down-side risks associated with fluctuation equal? That is, with a 50 percent chance to win under $100 or over $170, is either limit more likely than the other? An answer can be found in the "skewness" of the uninsured blackjack. Skewness equals -0.832, indicating that in a small number of tries, like 13 hands, the chance of going high is moderately greater than of going low.

So, should you take insurance when you have blackjack? No one garter fits all gams. Decide for yourself. But, at least, you now know the "cost" of insurance and the chance of various wins below or above the locked-in amount when you decline this side bet. As the immortal Sumner A Ingmark, the punters' poet, pondered:

My heart begins to race,
Whene'er my blackjacks face,
The dreaded dealer's ace.
Should I insurance place,
Or bigger winnings chase?