Insurance in Blackjack

Any of you who play Blackjack know that there is a bet called Insurance in the game.  By no means is this bet a ‘Piece of the Rock’.  For the few of you who are not familiar with Blackjack (if any) let me describe the bet to you.
Each player in the game of Blackjack receives two cards, sometimes face up and sometimes face down; it doesn’t matter.  The dealer receives two cards as well, but the dealer’s cards are dealt so that one is face down, the hole card, and one is face up.  If the dealer’s up card is an Ace, which counts as either one or eleven,  then there is a reasonable possibility that the hole card is a ten, thus giving the dealer a two-card 21, a Blackjack.  A Blackjack beats all other hands except another Blackjack, the latter case being a push.  Whenever the dealer’s up card is an Ace, the dealer offers the players the chance to ‘Insure’ their hand by making an Insurance bet.  This wager allows each player to place a wager of up to one half of his or her original bet on the proposition that the dealer will indeed have a Blackjack.  The bet pays 2 to 1.
In other words, a player taking this bet is wagering that the hole card will indeed be a ten.  For example, suppose the player’s original wager was $10 and the dealer’s up card is an Ace.  The player makes a $5 Insurance wager.  If the dealer’s hole card is a ten then the player loses the original $10 wager but wins 2 to 1 on the $5 Insurance bet ($10) and the whole thing is a push.  Well, there is an exception.  If the player insured a Blackjack and the dealer had a ten in the hole, then the original wager is a tie and the player wins $10.  If the dealer does not have a Blackjack, then the player loses the Insurance bet and play continues as usual.
Whoever thought up the word ‘Insurance’ for this bet certainly had a wry sense of humor.  Although there are times when the Insurance bet makes sense, and we will discuss this later, for the average player who is not counting cards this is a sucker bet.  Let’s see why.
To begin with let us look at the composition of a full deck of 52 cards.  There are 16 tens and 36 non-tens.  The ratio of non-tens to tens is 2.25.  This figure is the same for any number of decks.  So, if one were playing against an infinite deck, that is, a deck in which the ratio does not change with card removal, it is clear that the player’s expected return on the Insurance bet is:
exp = 2 x 16/52 – 36/52 = -0.076923 (1)
which is a house edge of around 7.69%.  As we will see, as the number of decks drops, the Insurance bet improves but still remains in favor of the house.
Consider 4 decks.  With an Ace up and two non-tens in the player’s hand, the remaining unseen cards consist of 64 tens and 141 non-tens.  In this case the expected return to the player is:
exp = 2 x 64/208 – 141/208 = -0.0625 (2)
which is a house edge of 6.25%.  How about one deck?  Well, with two non-tens in the player’s hand the calculation becomes
exp = 2 x 16/52 – 33/52 = -0.01923 (3)
or a house edge of 1.923%.  So you see, the wager is better with fewer decks but still favors the house.
In a game where the player’s cards are face up, the player has a Blackjack, and the dealer shows an Ace, it is quite common for dealers to try to cajole players into making an Insurance bet by saying, “Take even money, it’s a sure thing.”  Well, it is a sure thing but is it a good bet?  Let’s take a look.  In this case there are 15 tens left and 34 non-tens.  The expected return is:
exp = 2 x 15/52 – 34/52 = -0.076923 (4)
which is the same as the expected return in the infinite deck case.  Not a very good bet, is it?  If you check it out, you’ll see that with two tens in the player’s hand things are even worse (13.46% house edge).
Here is why this bet is so insidious and why I said earlier that whoever thought up the name had a wry sense of humor.  The name Insurance coupled with (some) dealer’s advice to “protect good hands” with this bet encourages players to take the worst of it.  What are good two-card hands?  Hands with tens in them. And as we have just seen, in general, these are the very worst hands to Insure.  In point of fact, the Insurance bet really is not about what cards the player holds.  The calculations above show that it is really a side bet about the ratio of non-tens to tens.  Whenever this ratio falls below 2, the Insurance bet is a good bet; otherwise it is a lousy bet.
Let me show you a simple counting system that one could employ to make perfect Insurance bets.  Assign each non-ten the number +1 and each ten the number -2.  Multiply the number -4 by the number of decks in the game and start the count there.  For example, if 4 decks are in play, the count starts at -16.  Then as cards are played, you add +1 for non-tens and subtract 2 for tens.  Whenever the total is greater than zero, insurance is a good bet.  In fact, the expected return on the insurance will be the count divided by the number of cards remaining in the deck.  It is fairly easy to see why.  If d represents the number of decks, then the count starts at -4d.  In the beginning there are 16d tens and 36d non-tens.  I can write the number -4d as
– 4d = 32d – 36d (5)
If t represents the number of tens seen and n represents the number of non-tens seen, then from (5) my running count is:
count =  -4d – 2t + n = 32d – 2t – 36d + n (6)
which can be rewritten as
count = 2(16d – t) – (36d – n) (7)
In words, (7) says that the count is just twice the number of tens remaining in the deck minus the number of non-tens remaining.  If we divide the count by the total number of cards remaining, therefore, we have twice the probability of a ten minus the probability of a non-ten.  This is just the expected return to the player (see calculations (1) – (4) above).
Now I don’t advocate using such a count; I present it only to make the point that tracking the deck can tell you a lot about when to make Insurance bets.  There are counting systems around (Vancura and Fuchs KO Count, for example) that are easy to use and correlate well with the perfect Insurance count above.  That is to say, they give you good information about Insurance as well as useful information about other aspects of the game.
I hope the message is clear.  Unless you are counting cards, don’t take insurance.  In particular, don’t buy in to the notion that Insurance “protects” a good hand.  This is a myth that casinos love to have you harbor.
Before I sign off here I should like to point out that if you have questions about any of my articles or anything else I can help you with, you can reach me directly via email by simply clicking on the word ‘Technigame’ in the ‘About the Author’ box below.  There is a link directly to my email and I would be happy to hear from you.  That’s it for now, see you next month.