What a Risk of Ruin Calculator Is Actually Computing

A risk of ruin calculator is one equation with three inputs.

risk of ruin = e ^ ( -2 x bankroll x edge per round / variance per round )

Every dropdown and slider you have ever used on one of these things is a front end for turning a rule set, a count system and a bet ramp into those three numbers, and then handing them to that exponential. The dropdowns are the hard part. The equation is not.

The question it answers is narrower than people assume. Playing this ramp, at this bet size, never resizing: what is the probability the bankroll touches zero at some point? That is all. Not how far down you will go. Not how bad a weekend can get. Not whether you finish the year ahead. One number about one catastrophic event.

Three things it assumes that you should not.

Bet size never changes as the bankroll moves, when in practice you would resize down, which lowers ruin and slows everything else.

The horizon is unbounded. The number is the chance of a zero touched at any point in a playing life with no end date, which makes it a ceiling rather than a forecast. Risk over any bounded stretch — this trip, this year, the next thousand hours — is strictly lower, and the shorter the stretch the lower it goes. A 5% lifetime figure is not a 5% chance of busting out this year.

And you are allowed to keep playing. Counting cards in your own head is legal in the United States, but a casino is private property and can bar you whenever it likes. A backoff ends the model's assumptions mid-sentence, and no calculator has a field for it.

The bankroll is denominated in bets, not dollars

Look at the exponent. Bankroll, edge and variance are all in the same denomination, so multiplying all of them by the same dollar figure changes nothing. Ruin depends on the ratio of bankroll to bet size and on nothing else.

So the honest unit is the minimum bet. On the ramp this article uses throughout — the Wizard of Odds' published six-deck Hi-Lo column: 1-to-10 spread, 4.5 of 6 decks dealt, Illustrious 18 plus Fab 4 — a 5% risk of ruin takes 782.5 units.

One caveat rides with every figure taken from that column, so take it here at the first one. Their simulated game is more liberal than the six-deck, dealer-stands-on-soft-17, DAS, 3:2, 75%-penetration game that carries a 0.41% house edge off the top. Every number in this article derived from that ramp is therefore an upper bound on the game you will actually sit down at. A worse game means a smaller edge, and a smaller edge means a bigger bankroll for the same risk.

782.5 units is $19,562 at a $25 table. It is the same 782.5 units and four times the cash at a $100 table, with exactly the same risk of ruin. Dollars are a display setting. Units are the physics. This is why counters describe a roll as "500 units" rather than in money: it is the only statement about bankroll that survives changing tables.

The fastest way to change your risk of ruin is usually not adding money, it is cutting the bet. Halving the unit doubles the number of units in the roll, and the next section is about what that does.

Doubling the bankroll does not halve ruin

Ruin is exponential in bankroll. Equal additions of bankroll produce equal multiplications of ruin, which is not how anyone's intuition works.

On that ramp, 782.5 units is a 5% risk of ruin. Double it to 1,565 units and ruin does not fall to 2.5%; it falls to about 0.25%, which is 5% squared. Go the other way, cut the roll to 391 units, and ruin is not 10%, it is about 22%, the square root of 5%.

The same structure explains why the last stretch of safety is so expensive. Getting from 5% to 1% costs 420 more units (1,202.9 minus 782.5). Getting from 1% down to 0.2% costs another 420. Each fixed block of bankroll divides your ruin by the same factor, forever, and never reaches zero. There is no bankroll at which this is safe. There is only a bankroll at which the risk is small enough that you have decided to stop caring, and that is a decision about you, not about blackjack.

The worked example

The reference ramp in full: six decks, 4.5 of the 6 dealt, a 1-to-10 spread, playing the Illustrious 18 plus the Fab 4, which is 22 indices in total. Quoted for that configuration: advantage 0.587%, average bet 1.68 units, standard deviation 2.27 units per round. Liberal simulated game, as above — upper bound.

Those three become the three inputs.

edge per round      = 0.00587 x 1.68  = 0.00986 units
variance per round  = 2.27 x 2.27     = 5.1529 units squared
2 x edge / variance                   = 0.003828 per unit of bankroll

Solve the exponential for the bankroll you want.

5% ruin:  -ln(0.05) = 2.9957  ->  2.9957 / 0.003828 =   782.5 units
1% ruin:  -ln(0.01) = 4.6052  ->  4.6052 / 0.003828 = 1,202.9 units

At a $25 table, 782.5 minimum bets is $19,562 and 1,202.9 minimum bets is $30,072.

Getting your own advantage, average bet and SD is the step this article is not going to walk you through, and no calculator walks you through it either — the dropdowns look them up. Those three are simulation outputs. Producing them means the edge at every true count for your exact rule set, how often each count occurs at your penetration, what your ramp bets at each of them, and then a few hundred million rounds dealt out to settle the averages. There is no napkin arithmetic that arrives at 0.587% and 2.27. Two honest options: take a published column whose rules, penetration and spread sit close to your table and treat the gap as error, or run the simulation. What you cannot do is take these three, which describe a liberal simulated game, and quietly assume they describe yours.

Which is why the configuration attached to those numbers is not decorative. At the same 1-to-10 spread, 4 of 6 decks dealt is an advantage of 0.368% and 5 of 6 is 0.837%. Required bankroll scales roughly with the reciprocal of the edge, so losing that half deck of penetration inflates the roll you need by about sixty percent, and gaining it cuts the requirement by about thirty percent. Penetration is worth more than most people think, and unlike the count or the ramp it is chosen at the door, by watching where the cut card goes in.

Swing against expectation

Over 100 rounds on that ramp, expectation is 100 x 0.00986 = 0.99 units. Standard deviation is 2.27 x the square root of 100 = 22.7 units. A swing-to-expectation ratio of 23 to 1.

Over an hour it is worse, because an hour is fewer rounds. Seventy hands an hour is the tempo of a six-deck table with four or five players; a genuinely full seven-spot table deals slower than that, which stretches every hour figure in this article. Expectation over 70 rounds is 0.69 units. Standard deviation is 2.27 x the square root of 70, or 19 units. Call it 27 to 1.

Expectation grows in proportion to hands played, and noise grows in proportion to the square root of hands played, so the ratio is 230.2 divided by the square root of the rounds: 27 to 1 over an hour, 23 to 1 over a hundred rounds, about 14 to 1 over a four-hour evening. It improves with volume, at the speed of a square root, and four hours in it is still 14 to 1. Almost everything unpleasant about advantage play follows from that.

It also identifies who actually goes broke. Bad players go broke from bad play, and their ruin is not interesting. The person this exponential is written for is playing the ramp correctly, taking every index, counting accurately, and is sitting inside a process where an hour of that work carries 27 units of noise for every unit of signal, and a whole evening of it still carries 14. Being right does not protect you from that. Capital does, and only in the exponential, diminishing way described above.

N0, and what "16% of the time" actually means

N0 is the number of hands at which cumulative expectation equals one standard deviation of the cumulative result. It is the square of the ratio you just computed per round:

N0 = (2.27 / 0.00986) squared = 52,960 hands

At 70 hands an hour, that is about 757 hours in a chair, and more than that if the tables you can find are fuller and slower. Not 757 hours of studying. Hours dealt to, at a real table, with the spread out.

At N0 the edge has grown to exactly one standard deviation, which means the expected result sits one sigma above zero. The chance you are still below zero at that point is the one-sigma tail: about 16%.

After 757 hours of correct play on that ramp, roughly one player in six is still behind. Not behind for the evening. Behind across all 52,960 hands. N0 is not the point where you break even. It is the point where the signal first becomes as large as the noise, which is a much weaker claim than it sounds like, and it arrives after the better part of a working year of table time. No calculator puts that on its front page.

What actually moves any of this

The rule set, first and by a distance. Six-deck, dealer stands on soft 17, DAS, 3:2, 75% penetration is a 0.41% house edge off the top. The same game with the dealer hitting soft 17 is 0.63%. The same game paying 6:5 costs another 1.36%, taking 0.41% to 1.77%. A skilled counter's realistic advantage is 0.5% to 1.5%, the Wizard of Odds' published band. You cannot spread your way across 1.77%. Risk of ruin at a 6:5 table is not a bankroll question.

Then the spread. Counting perfectly with no bet spread on that 0.41% game runs about -0.2%. Perfect basic strategy, perfect count, still losing. The count is an instruction to bet, not an ornament.

Then how often the count is worth acting on. Hi-Lo moves the edge roughly 0.5% per true count, with a band of 0.45 to 0.55. On the stand-on-soft-17 game, true count +2 is a player edge of +0.59%; on the hit-soft-17 version of the same game it is +0.37%. The edge crosses +1% at true count +2.8 on the stand game and +3.3 on the hit game. But rounds at or above +2 are 14.9% of all rounds, and rounds at or above +4 are 4.7%. The bankroll exists to survive the other 85%.

Even the most famous count-dependent decision is thinner than people expect. On six decks with 96 tens among 311 unseen cards, insurance wins 30.87% of the time and needs 33.33% to break even. The count tells you when that gap closes. Nothing about the gap is large.

And the games themselves are getting scarcer. On the Las Vegas Strip, 275 of 855 tables paid 3:2 as of September 2022 (Vegas Advantage), down from 720 in 2010. Zero single-deck 3:2 games remain in Las Vegas; the last one died at El Cortez on 28 April 2024. Any risk of ruin number is a statement about a specific game, and the specific games are thinning out.

So use the calculator in this order. Fix the rule set first, because it sets the edge — if you want the house edge for a placard you are standing in front of, thecutcard.com/table prices one for free. Pick the ruin you can live with, in units. Convert to dollars last, and convert again every time your minimum bet changes. Then look at N0 and decide whether 757 hours to reach the point where signal merely equals noise, with a 16% chance of still being behind when you get there, is a thing you want.

If you want the ramp, the 22 indices and the count drilled against a named rule set, with every edge figure on screen carrying the rules it was computed from, because 0.41% and 1.77% are the same game with one line changed, that is what The Cut Card is for; the basic strategy chart for whatever rule set you set is free and needs no account. It will not make the 23 to 1 any smaller. Nothing does.