Calculating insurance indices without simulators

Here's a method you can use that doesn't require the use of simulators. Although my description of this process is rather lengthy, with a little practice you can do the necessary calculations in 5-10 minutes. I'll use the Hi-Lo system and the single-deck game for this example, but this technique will work with just about any system and number of decks.

First, find the earliest point in the deck (or shoe) at which the insurance wager becomes a potentially profitable venture for the player. In the single-deck game, that point is reached when the first 5 cards dealt from the top of the deck are Non-tens. For the six-deck shoe, the earliest point occurs when the first 25 cards dealt from the top of the shoe are Non-tens (incredibly unlikely).

For the single-deck game, compute the average RC and TC for the first 5 cards when we know that they're all Non-tens *and we know that at least one of them is an ace (the dealer's up-card)*. How do we do that? We know that, along with the dealer's ace, the other four cards came from the 36-card set of Non-tens in the single deck. We set the dealer's Ace aside for the moment and concentrate on the remaining 35 cards. What is the average tag-value of each card in the 35-card subset? Using the tag-values for Hi-Lo, we add them up, then divide by 35:

4 + 4 + 4 + 4 + 4 + 0 + 0 + 0 - 3 = 17

17/35 = .4857

So the average tag-value of the cards in the subset of 35 Non-tens is .4857. We can use this figure to compute the average RC and TC of the first 5 cards dealt from the deck:

(4 * .4857) - 1 = .9428[RC]

.9428 * 52/47 = 1.0431[TC]

Notice that we've taken into account the 5th card (the dealer's Ace up-card), shown in the RC calculation with a tag-value of -1. Since this card is the only one of the five whose rank is known with certainty we can use its true tag-value in the calculations. Were it not for the presence of at least one ace (the dealer's up-card), the dealer wouldn't even offer the insurance wager. There may or may not be additional aces appearing in the other 4 cards; their occasional presence there was taken into account when we computed the average tag-value (.4857).

Let's compute the TC for the next two potentially profitable insurance scenarios:

When the first 6 cards dealt from the deck are Non-tens:

(5 * .4857) - 1 = 1.4285[RC]

1.4285 * 52/46 = 1.6148[TC]

When the first 7 cards dealt from the deck are Non-tens:

((6 * .4857) - 1 = 1.9142[RC]

1.9142 * 52/45 = 2.2120[TC]

Now, repeat these calculations at three consecutive points somewhere around the maximum penetration you expect to be able to play. I'll use a few points at mid-deck for this example:

When 26 cards have been dealt and no more than 7 of them are Tens:

(18 * .4857) - 7 - 1 = .7426[RC]

.7426 * 52/26 = 1.4852[TC]

When 27 cards have been dealt and no more than 7 of them are Tens:

(19 * .4857) - 7 - 1 = 1.2283[RC]

1.2283 * 52/25 = 2.5549[TC]

When 28 cards have been played and no more than 7 of them are Tens:

(20 * .4857) - 7 - 1 = 1.714[RC]

1.714 * 52/24 = 3.714[TC]

I determined that the maximum number of Tens allowable in the three subsets of played cards was 7 by looking at each subset's associated number *of unplayed cards*. With 28 played cards, there is a subset of 24 unplayed cards. Looking at those 24 **unplayed** cards, insurance is profitable when at least 9 of the 24 are Tens. Subtracting those 9 from the original 16 Tens in the deck leaves us with a maximum of 7 cards in the 28 **played** cards.

You now have 6 TCs calculated at opposite ends of the playable portion of the deck. You could, if you felt it necessary, add all 6 TCs together, then divide the total by 6 to get an average TC:

1.0431 + 1.6148 + 2.2120 + 1.0804 + 1.4852 + 2.5549 + 3.714 = 13.7044

13.7044 / 6 = 2.2841

Round off the 2.2841 to +2 for your Hi-Lo, single-deck insurance index.

If you care to bother with it, you can perform these calculations at several other points throughout the deck. You'll find the TCs slowly rising (in groups of three) the deeper you go into the deck. The biggest reason for this is the decreasing denominator in the calculations to convert RC to TC. For example, going from 52/47 to 52/46 isn't nearly as big a leap as going from 52/25 to 52/24.