Flooring vs Rounding the True Count, Measured

Ask what a true count of +1 is worth and the honest answer starts with a question back: floored, truncated or rounded? A "+1" is not one number. It is a bucket of raw true counts whose edges depend on the rounding convention.

To see how much that matters, we ran 62,094,739 rounds of simulated blackjack, and before every round computed one raw value: the Hi-Lo running count divided by the exact number of decks remaining. That raw true count was turned into a whole number three ways, and every round was recorded under all three. The rounds are identical. Only the labels differ.

The result is three scoreboards for one game, and the differences come down to two mechanics: a truncated zero bucket two points wide, and a half count offset between floored and rounded labels.

The run

The rules match the larger simulation behind our pieces on how the player's edge moves with the true count and how often each true count comes up: six decks, dealer stands on soft 17 and peeks, blackjack pays 3:2, double on any two cards, double after split, split to four hands, one card to split aces with no resplitting, no surrender, a cut card at 75 percent penetration, one player, a flat bet of one unit, and basic strategy from the app's 4 to 8 deck chart with no count based plays. This measurement used a separate script.

The three conversions are the ones our piece on rounding conventions defines. Floor always takes the next lower whole number: +1.7 becomes +1 and -0.3 becomes -1. Truncate drops the fraction toward zero: -0.3 becomes 0 and -1.7 becomes -1. Round goes to the nearest whole number with exact halves rounding up: +1.5 becomes +2 and -0.5 becomes 0.

One design choice keeps the comparison clean: bets were flat and no playing decision depended on the count, so the convention changed no outcomes, only the label attached beforehand.

How often each label comes up

Truncation's zero row stands out: 45.48 percent of all rounds, against 26.61 floored and 26.24 rounded. A truncated zero holds every raw count from just above -1 to just under +1, a band two full points wide, while a floored zero holds 0 up to just under +1 and a rounded zero holds -0.5 up to just under +0.5. Same rounds, wider bucket.

Above zero the floored and truncated columns are identical row for row, because on positive numbers down and toward zero are the same direction. They disagree only below zero.

Rounding pulls every raw count from +0.5 upward into the positive labels, so rounds labeled +1 or higher come to 35.90 percent rounded against 26.53 floored or truncated, and +2 or higher comes to 19.78 against 14.77. The rounds themselves are the same; only the sorting changed.

What each label is worth

The second table gives the average result per round, in percent of the initial bet, with the standard errors in one column.

The quiet finding: the same label is worth a different amount under each convention. At +1 the rounded average is +0.13 against +0.31 floored, a gap of 0.18 points. At +2 it is +0.51 against +0.71, a gap of 0.20. At +3 it is +1.05 against +1.25, again 0.20. At +4 and +5, where the standard errors are larger, the gaps are 0.38 and 0.09.

The reason is bucket position. A floored +1 holds raw counts from +1 up to just under +2, and a rounded +1 holds +0.5 up to just under +1.5, so the floored bucket sits about half a count higher. Our edge by true count piece measured the average result rising about 0.5 percent per true count from -3 to +3 in this rule set. Half a count at that slope is about a quarter point of expectation, and the gaps of 0.18 to 0.20 at +1 through +3 are consistent with that.

Under all three conventions the shape near zero is the same: the 0 label averages below zero, and the first label with an average above zero is +1. But how far above zero +1 sits depends on the convention, +0.31 floored and truncated against +0.13 rounded.

Below zero, the rows shift by one

Compare the truncated -1 row with the floored -2 row: 12.41 percent of rounds averaging -1.16, against 12.43 percent averaging -1.17. Compare truncated -2 with floored -3: 6.82 percent at -1.70 against 6.75 at -1.70. Below zero, truncation moves every raw count that is not a whole number up exactly one label, so the truncated negative rows are near copies of the floored rows one step down; they differ only where a raw count lands exactly on a whole number, which is rare.

A negative count quoted without its convention is ambiguous: "-1" covers 18.96 percent of rounds floored and 12.41 truncated, and those buckets share only the rounds at exactly -1.

Does this run agree with the bigger one?

Our published 414 million round run floored the count, so this run's floored column is the one to compare. The shares agree almost exactly: 26.62 percent at zero and 11.78 at +1 there, against 26.61 and 11.76 here.

The averages agree at zero and +1: -0.176 and +0.341 there against -0.16 and +0.31 here, gaps of about 0.02 and 0.03 points, inside this run's standard errors of 0.03 and 0.04. At +2 they do not: +0.858 there against +0.71 here, a gap of about 0.15 points, about 2.3 times the two runs' standard errors combined (0.06 here, 0.022 there). That is more than sampling error usually explains for a single row. It is the only row from -5 through +5 with a gap that large, and with eleven rows compared one such gap can turn up by chance; the bigger run's +0.858 is the more precise figure.

All standard errors here treat rounds as independent, but rounds from the same shoe are correlated, so the real uncertainty is somewhat larger than the quoted figures. Throughout, differences of a few hundredths should be read as ties.

A count is only meaningful with its convention

The practical lesson is bookkeeping, not betting. Any true count figure is the product of a raw count and a rounding rule, and comparing two figures produced under different rules compares mismatched buckets. Before quoting or comparing a count figure, check which convention produced it.

Index numbers carry the same dependency: they come from simulations that had to assume a conversion method. Use the method your index set was generated with, and if the source does not say, pick one and treat the choice as part of the index set.

One more gap separates this from a live table: dividing by the exact decks remaining is a simulation convenience, while a player estimates decks from the discard tray, the skill our piece on deck estimation practice covers.

And the standing cautions apply. Counting with your head is legal in the United States, but casinos are private businesses that may refuse service to suspected counters, and no technique removes risk from the game. Nothing here is betting advice; these are measurements of labels.

Frequently asked questions

Is it better to floor or round the true count?

This run cannot answer that directly: every round was bet flat with no count based plays, so the convention changed no outcomes. It measures what the labels mean. Our rounding conventions article cites simulation software documentation reporting that flooring and rounding perform similarly and truncating is slightly weaker, so pick one method, ideally the one behind your index numbers, and never mix.

Why does truncating give so many zero true counts?

Because a truncated zero is a band two points wide. It holds every raw true count from just above -1 to just under +1, while a floored zero holds 0 up to just under +1 and a rounded zero holds -0.5 up to just under +0.5. In our run the truncated zero row held 45.48 percent of rounds, against 26.61 floored and 26.24 rounded, from exactly the same rounds.

Is a true count of +2 worth the same under every rounding convention?

No. In our run the +2 label averaged +0.71 percent of the initial bet when floored and +0.51 when rounded, a gap of 0.20 points. A floored +2 holds raw counts from +2 up to just under +3, and a rounded +2 holds +1.5 up to just under +2.5, so the floored bucket sits about half a count higher. Gaps of 0.18 and 0.20 appear at +1 and +3.

Why do flooring and truncating give the same count on positive numbers?

Above zero, flooring moves down and truncating moves toward zero, the same direction, so +1.7 becomes +1 under both. They disagree only below zero, where truncation moves every raw count that is not a whole number up one label: the truncated -1 row in our run, 12.41 percent averaging -1.16, nearly duplicates the floored -2 row, 12.43 percent averaging -1.17.

Which rounding convention were published index numbers generated with?

It depends on the source, and this run cannot tell you. Index numbers come from simulations that had to assume a conversion method, so the ideal is to match it. If the source documents its method, follow it; if not, choose floor, truncate or round once and treat the choice as part of the index set, because consistency matters more than the specific method.

Keep reading

• Rounding the True Count: Floor, Truncate or Round
• How Often Each True Count Comes Up
• How the Player's Edge Moves With the True Count