Insurance in Blackjack
Whenever dealer is dealt an up card of ace in most games it is customary for insurance to be offered to the player for his 2 card hand. If player accepts he makes a side wager of 1/2 of his main bet before dealer checks to see if he has blackjack (ten or picture card in the hole.) If it turns out that dealer has blackjack then player is paid 2 to 1 on this side bet. If it turns out dealer doesn't have blackjack then player loses this side bet. This side bet in no way affects how his main bet is handled. So the question is, "Should player take insurance?"
Basically this is a very simple problem. When to take insurance depends on how likely it is for dealer's hole card to be a ten value card (ten/picture.) Whenever the probability of a ten value card being drawn is more than 1/3 = 33.3333....33% then the 2 to 1 odds payoff make insurance a positive expectation bet.
Determining when probability of a ten value card is greater than 1/3
Any counting system that groups cards into 2 (and only 2) groups of cards consisting of all ten value cards in one group and all non-ten value cards in the other is capable of determining when probability of drawing a ten value card is greater than 1/3 if implemented perfectly. However, at the table it is not very practical to expect to implement a system perfectly if the mechanics are too messy. There is one system that both works perfectly relative to insurance and is also simple enough. Non-tens are tagged -1 and tens tagged +2. For a full shoe there are 36 x decks non-tens and 16 x decks tens. The starting count of a full shoe is 36 x decks x (-1) + 16 x decks x (+2) = (-4) x decks. Every time a non-ten is dealt 1 is added to the current count. Every time a ten is dealt 2 is subtracted from current count. Whenever current count equals 0, probability of drawing a ten value card equals 1/3, at which point the insurance side bet is an even proposition. Whenever count is negative the insurance bet has negative expectation. Whenever count is positive the insurance bet has a positive expectation. We'll call this the insurance count because it can perfectly predict when insurance is and is not a positive expectation bet.
Using the insurance count as a general purpose count
Although the insurance count is perfect for insurance it is not so good at evaluating when player's expectation is positive in general. Insurance opportunities arise only when dealer's up card is an ace and even if up card is an ace it still may not be a positive expectation opportunity. Also the insurance bet is limited to 1/2 of the main bet. If somehow a player was allowed to wager any amount he wanted on insurance then that would make the insurance count a much more powerful weapon but unfortunateky this is not the case. Therefore the gain in EV for insurance using the insurance count is outweighed by the fact that its general evaluation abilities aren't very good. A better choice is to use a count that is better in general evaluation, although such a count won't be perfect relative to insurance.
Deciding when to insure using a count that is not perfect relative to insurance
One way to decide when to insure using a count is through simulation. The result is a single reference that applies to all conditions for a given counting system and a given number of decks. However this website is about computing first before resorting to simulation. The parameters needed to compute whether or not an insurance opportunity is positive expectation are the "dumb animal count" and number of cards remaining to be dealt. Additionally information about other specifically removed cards can be included for even more accuracy. By "dumb animal count" I am alluding to the fact that basically all a counter knows is a number known as running count plus an estimation of number of decks remaining. From this information a number known as "true count" can be calculated. True count equals running count divided by number of remaining decks and it represents the imbalance per deck of the groups of ranks that define a given counting system. For example the HiLo counting system consists of 3 groups:
- 2,3,4,5,6 --- low cards
- 7,8,9 --- medium cards (a.k.a neutral cards)
- T,J,Q,K,Ace --- high cards
A positive running count means that there are more cards from the high card group than the low card group remaining to be dealt while a negative running count means more low than high remain. The true count measures the relative strength of this imbalance on a "per deck" basis. Since for HiLo there are an equal number of high and low cards present for a full shoe, if count is positive this means that the probability of drawing a T,J,Q,K on the next card tends to be increased. Whenever the probability of drawing a T,J,Q,K is greater than 1/3 then taking insurance yields a positive expectation. It is possible to compute how many cards must be remaining to be dealt for a given running count to yield a positive insurance expectation. Of course some running counts may never yield a positive expectation while some always do. There is no better way to decide when to insure using a counting system that by its nature is not perfect relative to insurance than to enumerate all of the insurance expectation reference points in this way. The resulting data is certainly far more than is needed but is included for completeness. Below are links to insurance data for both the HiLo counting system as well as the KO system for 1, 2, 4, 6, and 8 decks. The data comes from a program capable of considering all possible subsets of any given counting system at any given penetration (cards remaining.) Additionally it can allow for any number of specific removals. For insurance it is a given that an ace must be specifically removed to account for the necessity of dealer's up card being an ace for insurance to be an option. Each of player's 2 cards relative to the counting system being used are also removed. From this input the program is able to compute the probability of drawing a ten value card which is all that is needed to determine whether an insurance bet has a positive expectation. The data takes into consideration the counting system's "dumb animal count" which has already considered the composition of the 2 card hand relative to the counting system versus dealer's up card of ace at the time of the insurance opportunity. It would be possible to generate similar data for any counting system although some of the more complex systems would require more computing power than I have, especially for more decks.
At the end of each set of data there is a true count approximation for each 2 card hand relative to either HiLo or KO. All I did was somewhat arbitrarily estimate an approximate single value true count for each 2 card hand to apply to all penetrations. The best way would be to follow the data exactly but a good estimate would get most of the gain without being so taxing. The insurance index output by a simulation is a single count value that does not take into consideration player's hand composition and is an average over all penetrations (cards remaining to be dealt.) It would be possible to output data that is more comparable to a sim's output by ignoring hand composition and listing data relative to cards remaining. Then a single figure would be estimated from this data. Although this would be more accurate than a sim I chose to include each 2 card hand as addtional information that otherwise wouldn't be available. The greater the number of decks, the less effect a player hand's composition has.