Blackjack insurance

"Insurance" is a blackjack side bet, offered to the player when the dealer has an ace as his upcard. The player must make an additional wager equal to half of his original bet, and if the dealer does go on to make blackjack, the insurance bet is paid at 2 to 1, resulting overall in a push.

Insurance odds

The "odds" of any bet is the ratio of the number ways of winning set against the number of ways of losing - see the house edge article; in order for a bet to be "fair", the payout has to match the odds. For example: the odds on a coin-toss are 1 to 1 - heads/tails or tails/heads - so in this case the payout must be equal to the size of the bet - 1 to 1 - if the bet is to be fair and for neither side to have the edge; the odds on a single roulette number are 36 to 1, so a 36 to 1 payout is required for a fair bet. And so on.

The payout on a successful insurance bet in blackjack is 2 to 1; let's have a look at the odds and see how closely the payout matches them.

Out of the 52-card deck, there are three "seen" cards - the dealer's ace plus the player's two cards - and 49 "unseen". Assuming the player doesn't have a 10 value card in his hand, there are 16 ten value cards remaining elsewhere in the deck, and therefore 33 non-tens. 33 to 16 works out at odds of 2.0625 to 1 (33 divided by 16), so it's apparent that the actual payout we receive, 2 to 1, does not match the true odds of the bet. In order to justify taking insurance in these circumstances we'd need to be offered a 2.0625 to 1 payout, or \$20.625 on a \$10 bet. The \$20 payout we actually receive is not sufficient.

Things rapidly get worse as we start to add 10-value cards to our hand: assuming the player has one ten card and a non-ten, there are now only 15 tens remaining and 34 others, giving us odds of 34 to 15, or 2.26 to 1. The true odds have deteriorated further with the absence of just one more of those cards that would give the dealer his blackjack and win the insurance bet for us; now we'd need a payout of \$22.60 on a \$10 bet to justify taking insurance.

In the worst possible scenario, if the player has two tens in his hand the odds now stand at 35 to 14, or 2.5 to 1, on winning the insurance bet. A \$10 bet now requires a \$25 insurance payout to justify the wager - \$20 at 2 to 1 definitely doesn't cut it!

Ironically enough, those hands which a player would be least likely to buy insurance for of the three categories - those small non-ten hands such as 5/2, 9/6, 8/7 etc - are just those hands whose actual odds are closest to the 2 to 1 payout (at 2.0625 to 1) and as such would make it the "least" unjustified bet. It's still a rotten bet, make no mistake, but the odds aren't as bad as they are for the one-ten hand at 2.26 to 1 - or the ten/ten hand at 2.5 to 1. Players frequently insure their blackjacks and 10/10 hands, but rarely their 5/6 or their 8/2. This is one of those situations where the most un-intuitive play is the also the least incorrect one.

Expectation of the insurance bet

In a six-deck game, assuming there are no tens in the player's hand, there are 96 tens out of the total of 309 unseen cards. This gives us the following calculation for the expectation of the bet:

 (2 × (96/309)) + (-1 × (213/309)) = - 0.0679

The expectation is -0.0679, or a house edge of of 6.79%.

The following table lists the house edges for various decks and the number of visible ten cards:

 Number of 10 cards 1 deck 2 decks 4 decks 6 decks 8 decks None 2.04% 4.95% 6.34% 6.79% 7.02% 1 8.16% 7.92% 7.8% 7.76% 7.74% 2 14.29% 10.89% 9.26% 8.74% 8.47%

"Even money"

Since one of the most common insurance practices is to insure a blackjack and take a guaranteed win, rather than go for the 3 to 2 payout and risk a push, let's have a look at the percentages and see what we actually sacrifice in overall money terms when we accept the dealer's offer of "even money":

Again, out of the 52-card deck there are 49 unseen cards, of which 15 are tens, the player having one of them in his blackjack. Assuming he takes "even money" he'll win one unit every time, whether or not the dealer has blackjack, for a total of 49 units. If he ignores the insurance option and goes for the 50% extra on his blackjack when the dealer doesn't have one himself, he wins 1.5 units on those 34 occasions when the dealer misses, and pushes the remainder. 1.5 × 34 = 51. So, buy not taking even money he averages 51 units, and by taking it he wins 49 - two units less. 2 out of 51 is 3.9%, so by taking even money we give away 3.9% of the value of our hand. Based on a \$10 bet, we sacrifice an average 39 cents. \$100 bet, and we lose an average \$3.90.

In fact, "insurance" is not insurance at all - using such a respectable financial commodity to describe this bet is just one example of the many neat little tricks casinos play to encourage us to part with our money quicker. All it is is a side bet on whether or not we think the dealer has a blackjack - it has nothing whatsoever to do with our original bet, which wins, pushes or loses irrespective of the insurance bet.

And since we know that the payout on the bet doesn't match the actual odds, we never take insurance.

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