What a Card Is Worth

Every card that leaves the shoe changes the game behind it. Not by much — a single card out of 312 moves the house edge by a few hundredths of a percentage point — but the direction is never random, and the sizes are exactly computable. That calculation is what card counting is: the whole apparatus of running counts and true counts is a way of adding those hundredths up in your head.

This page derives the numbers underneath it. What one card of each rank is worth, why three whole numbers approximate that well enough to be useful, and the exact count at which each of the contested hands stops being played the way the chart says.

Effect of removal

Take one card out of a fresh 6-deck shoe and recompute the whole game. The bars below are the difference: how far the house edge moves, in percentage points, when that one card is gone. Bars above the line are cards you want gone.

The 5 is the card the dealer most wants to keep: removing one lowers the house edge by +0.124 points, more than the 4 at +0.097. At the other end, removing an ace costs you 0.095 points, just ahead of the 10. The shape is the reason a counter raises their bet when small cards are gone: low cards make the dealer’s hand, because a dealer who must draw to 17 is rescued by a 5 and busted by a ten.

Checked against two published tables

These are not numbers worth taking on trust, so they are not asked to be. The same calculation run at one deck and at 6 lands against two independent published tables, computed decades apart by different people with different engines.

At 6 decks the two series correlate at 0.9977, with an average gap of 0.0077 points and a worst case of 0.0172. At one deck, where the numbers are roughly six times larger, the correlation is 0.9990. The residual is the one approximation left in the engine, described at the bottom of this page — it shrinks every figure slightly toward zero without changing the order of any of them.

Why three whole numbers are enough

Hi-Lo, the system almost everyone learns first, throws nearly all of that detail away. Low cards count +1, high cards −1, the 7, 8 and 9 count nothing. Set against the real effects, those tags look crude — and they are. The column on the right is what each tag would be if it were allowed to be a decimal, scaled so the whole set lines up as closely as possible with the effects above.

Two things fall out of that column. The tag Hi-Lo gets furthest wrong is the 7: counted as nothing, worth +0.49 — about half a tag, rounded to zero. And the 2 and the 5 are given the same weight although the 5 is worth 1.9× as much; systems that split that difference exist, and they are harder to run at speed.

And yet. The betting correlation between those three integers and the real per-card effects — weighted by how many of each rank a deck holds, so the tens count four times — is 0.9636. Computed instead from the published effect-of-removal table rather than ours, it is 0.9634. That agreement matters more than the figure itself: it is a claim about the ratios between ranks, which the engine’s remaining approximation cannot touch.

Insurance, exactly

Insurance is the one bet on the table that needs no engine at all. It pays 2:1 that the dealer’s hole card is a ten, so it is profitable exactly when tens make up more than a third of what is left. With the dealer’s ace face up and the count at neutral, tens are 30.9% of the rest of the shoe — short of the 33.3% it would take, which is why the book answer is never take it.

Tens cross a third at a true count of +3.13 with 4 decks still to come, so the index is +3 — the number Schlesinger publishes. The decimal is not a constant, though, and it would be sleight of hand to print it as one: the crossover slides from +2.50 deep in the shoe to +3.17 early in it. Every one of those still rounds to +3, which is what makes the published index a safe thing to memorise and the decimal a bad thing to quote on its own.

Where the count changes your play

A basic strategy chart is the right answer at a neutral shoe. Shift the composition far enough and the second-best action overtakes the best one; the true count where that happens is the hand’s index number. Below is that crossover for the 29 of the 31 contested hands the count can settle, solved for rather than looked up, measured with 4 of the 6 decks still to come. A true count is a per-deck figure precisely so that it does not depend on how deep the shoe is, and the generator checks that: every index below is unchanged, to within one, a deck further in.

The striking part is the top of the table: 16 of these 29 flip at a count of even or below, and 9 sit within a single true count of neutral. Counting is popularly imagined as something that pays off when the shoe goes rich; most of these hands change because it went poor.

Two of the 31 are missing from that table: Pair of 3s against 8 and Pair of 7s against 8 never change at any count inside ±12. That is a weaker claim than it looks, and it is worth saying why: these indexes are computed for a player total, the convention the published tables use, which means the two cards in your own hand are left in the shoe. For a pair that is a fiction — a pair of 3s really is two 3s gone — and taking them out instead makes both of these flip, at around +3 and +5. The published convention is the one that matches Schlesinger where he can be checked, so it is the one used here; but for these two hands the answer depends on the convention rather than on the shoe, and the right thing to report is that rather than a confident never.

Close decisions flip first

Rank those hands by how narrowly the best action wins at a neutral shoe, and rank them again by how far the count has to move to overturn it, and the two orders agree at a Spearman correlation of 0.76. That is the same measurement the close-calls pages are curated on, seen from the other side: a decision that basic strategy barely wins is a decision a thin shoe can take away. Hard 16 against a ten — the tightest call on the whole chart, worth 0.06% of a bet — turns over at +0.44. Hard 16 against an ace, which basic strategy wins by 250× as much, needs +8.74.

Against the Illustrious 18

The best-known list of index plays is Don Schlesinger’s Illustrious 18, the eighteen deviations that return the most to a Hi-Lo counter. 9 of them are among the contested hands here, and all 17 playing indexes were recomputed to check this page’s method against his published figures: 14 of 17 land on the same whole number, and 17 of 17 land within one. The insurance index above is the eighteenth.

What this model does not do

The engine deals the first three cards of a hand — both of yours and the dealer’s up-card — without replacement, then holds the remaining composition fixed while the hand plays out. Depletion within a hand is not modelled. That approximation is why every effect-of-removal figure above sits slightly closer to zero than the published ones, and it is why this page carries no house-edge-by-deck-count table: the deck-count effect this model produces runs about 30% steeper than the published one, which is accurate enough to see the shape and not accurate enough to print.

Nothing here is betting advice, and nothing here is a system. It is the arithmetic that systems approximate, computed once and written down. The dataset behind this page is downloadable, along with what each figure was checked against.