Blackjack Variance
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A caveat is in order: these return to player percentages are based upon long-term statistical probabilities and there is significant variance in results over the short term. Blackjack statistics ‘teach’ players what to do in certain situations. For example, if you are holding a 12-value hand, should you hit, stand or, double? Jan 13, 2018 Free Blackjack Photos LeoVegas: Up to £400 in bonuses + 100 bonus spins 18+ New eligible UK players only. Select Casino offer on Free Blackjack Photos sign-up and deposit. 4 deposits of £10, £20, £50, £100 matched with a bonus cash offer of same value (14 day expiry). Jun 11, 2019 Example 1: A card counter perceives a 1% advantage at the given count. From my Game Comparison Guide, we see the standard deviation of blackjack is 1.15 (which can vary according to the both the rules and the count). If the standard deviation is 1.15, then the variance is 1.15 2 = 1.3225. The portion of bankroll to bet is 0.01 / 1.3225 = 0.76%.
Standard Deviation is a measure of how results are distributed within a range of possible outcomes. This score is useful when comparing averages – for example two scores may have the same average of ‘50’ with one comprising of results entirely between 45 and 55 and the other having results ranging from 1 through to 100. The second set of scores is more widely distributed than the first, which will be reflected in a higher standard deviation score.
In Blackjack the standard deviation can be used to show you what the probability of winning or losing a number of betting units is, based on the number of hands you play. We could use this score to answer questions such as ‘what is the probability of winning more than 20 betting units over the course of 100 hands?’ or “Is winning 30 bets over 200 hands a normal outcome, or did I get lucky?’
This article will show you how to calculate (with the help of a calculator) your standard deviation in Blackjack and how to use this information to your advantage, whether you play live or in an online casino.
Calculating The Standard Deviation In Blackjack – A Note About Distribution Curves
You get to the probability of an event occurring by comparing your standard distribution score to a ‘bell-curve’ of possible outcomes, known as a ‘normal distribution’. The score is calculated in such a way as to show that 68% of outcomes will fall within 1 standard deviation of your average, 95% of outcomes fall within 2 standard deviations and 99.7% of outcomes fall within 3 standard deviations.
In other words, once we have worked out what the standard deviation score is for a certain set of data we then compare this to the normal distribution curve to arrive at a probability of any single score being within the normal set of expected outcomes.
Blackjack Standard Deviation – How Can We Chart The Distribution Of Blackjack Outcomes?
Outcomes of a single blackjack hand can be mapped precisely – since we know the rules, probabilities and all of the cards in the deck. The most common outcome is to win or lose one betting unit, with splits, doubles and blackjack making it possible to win or lose more. With the larger number of single units dominating it is possible to work out that the standard deviation for a single hand of 6-deck Blackjack is exactly 1.1418 based on card distributions and rules alone.
| Net Win in Blackjack | |||
| Net win | Total | Probability | Return |
| 8 | 1079 | 0.00000063 | 0.00000506 |
| 7 | 10440 | 0.00000612 | 0.00004287 |
| 6 | 64099 | 0.00003761 | 0.00022563 |
| 5 | 247638 | 0.00014528 | 0.00072642 |
| 4 | 1307719 | 0.00076721 | 0.00306885 |
| 3 | 4437365 | 0.00260331 | 0.00780994 |
| 2 | 99686181 | 0.05848386 | 0.11696773 |
| 1.5 | 77147473 | 0.04526086 | 0.06789129 |
| 1 | 540233094 | 0.31694382 | 0.31694382 |
| 0 | 144520347 | 0.08478716 | 0 |
| -0.5 | 76163623 | 0.04468366 | -0.02234183 |
| -1 | 684733650 | 0.40171937 | -0.40171937 |
| -2 | 71380000 | 0.0418772 | -0.0837544 |
| -3 | 3559202 | 0.00208811 | -0.00626434 |
| -4 | 828010 | 0.00048578 | -0.00194311 |
| -5 | 152687 | 0.00008958 | -0.00044789 |
| -6 | 30536 | 0.00001791 | -0.00010749 |
| -7 | 3972 | 0.00000233 | -0.00001631 |
| -8 | 305 | 0.00000018 | -0.00000143 |
| Total | 1704507420 | 1 | -0.00291455 |
This table reflects a standard deviation of 1.1418.
By applying this number directly to a ‘normal distribution’ – or bell curve - we find that over an infinite sample, in a single hand of blackjack you will win or lose 1.1418 betting units or less 68% of the time, win or lose 2 standard deviations or 2.2836 betting units or less 95% of the time and win or lose 3 standard deviations or 3.4254 units or less 99.7% of the time.
While this score is interesting, the application of it benefits from adding a second variable – the number of hands played.
Standard Deviation In Blackjack – Using The Information To Predict Win / Loss Runs
Finally we get to the key practical application of working out standard deviations in blackjack games – assessing the likelihood of winning or losing certain amounts of units over specified numbers of hands. Here is the formula to work this out based on your hand sample size:
(Square Root Of The Number Of Hands Played)*1.1418
Here are some working examples:
100 hands played, square root = 10 * 1.1418 = 11.418
This shows that 68% of the time you will win or lose 11.418 units or less over the course of 100 blackjack hands, 95% of the time you will fall within 2 standard distributions and win or lose less than 22.836 units – while 99.7% of the time your outcome over 100 hands will be within 3 standard deviations, or + / - 34.2 units.
300 hands played, square root = 17.32 * 1.1418 = 19.77
Here the distribution of blackjack outcomes predicts you will win or lose 19.77 units 68% of the time you play 300 hands, 95% of the time you will fall within 2 standard deviations and win or lose 39.54 units and 99.7% of the time you will fall within 3 standard distributions and win or lose < 59.31 units.
As you can see, the higher the number of hands played the smaller the relative impact of chance. Of course you also need to take into account the house edge of 0.05% or so when making these calculations!
Do not worry if you do not have a pocket calculator with you at the casino, it is straight forward to work out the blackjack standard deviations for different sized sessions in advance and gain an insight into how the average distribution of outcomes affects your chances of either winning or losing certain amounts of cash. Once you get an idea of the types of swings which are normal in the game you will feel more comfortable at the blackjack tables, whether live or online!
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Introduction
This appendix presents information pertinent to the standard deviation in blackjack. It assumes the player is following basic strategy in a cut card game. Each table is the product of a separate simulation of about ten billion hands played.
As a reminder, if the variance of one hand is v, the covariance is c, and the number of hands played at once is n, then the total variance is n×v + n×(n-1)×c.
The following table is the product of many simulations and a lot of programming work. It shows the variance and covariance for various sets of rules.
Blackjack Variance Chart
Summary Table
| Decks | Soft 17 | Double After Split | Surrender Allowed | Re-split Aces Allowed | Expected Value | Variance | Covariance |
|---|---|---|---|---|---|---|---|
| 6 | Stand | Yes | Yes | Yes | -0.00281 | 1.303 | 0.479 |
| 6 | Stand | No | No | No | -0.00573 | 1.295 | 0.478 |
| 6 | Hit | Yes | Yes | Yes | -0.00473 | 1.312 | 0.487 |
| 6 | Hit | No | No | No | -0.00787 | 1.308 | 0.488 |
| 6 | Hit | Yes | No | No | -0.00628 | 1.346 | 0.499 |
| 6 | Hit | No | Yes | No | -0.00699 | 1.272 | 0.475 |
| 6 | Hit | No | No | Yes | -0.00717 | 1.311 | 0.488 |
| 8 | Hit | No | No | No | -0.00812 | 1.309 | 0.489 |
| 2 | Hit | Yes | No | No | -0.00398 | 1.341 | 0.495 |
By way of comparison, Stanford Wong, in his book Professional Blackjack (page 203) says the variance is 1.28 and the covariance 0.47 for his Benchmark Rules, which are six decks, dealer stands on soft 17, no double after split, no re-splitting aces, no surrender. The second row of my table shows that for the same rules I get 1.295 and 0.478 respectively, which is close enough for me.
Effect on Variance of Rule Changes
The next table shows the effect on the expected value, variance and covariance of various rule changes compared to the Wong Benchmark Rules.
Effect of Rule Variation
| Rule | Expected Value | Variance | Covariance |
|---|---|---|---|
| Stand on soft 17 | 0.00191 | -0.00838 | -0.00764 |
| Double after split allowed | 0.00159 | 0.03753 | 0.01091 |
| Surrender allowed | 0.00088 | -0.03629 | -0.01247 |
| Re-split aces allowed | 0.00070 | 0.00207 | 0.00037 |
| Eight decks | -0.00025 | 0.00071 | 0.00063 |
| Two decks | 0.00230 | -0.00530 | -0.00422 |
What follows are tables showing the probability of the net win for one to three hands under the Liberal Strip Rules, defined above.
Liberal Strip Rules — Playing One Hand at a Time
The first table shows the probability of each net outcome playing a single hand under what I call 'liberal strip rules,' which are as follows:
- Six decks
- Dealer stands on soft 17 (S17)
- Double on any first two cards (DA2)
- Double after split allowed (DAS)
- Late surrender allowed (LS)
- Re-split aces allowed (RSA)
- Player may re-split up to three times (P3X)
6 Decks S17 DA2 DAS LS RSA P3X — One Hand
| Net win | Probability | Return |
|---|---|---|
| -8 | 0.00000019 | -0.00000154 |
| -7 | 0.00000235 | -0.00001643 |
| -6 | 0.00001785 | -0.00010709 |
| -5 | 0.00008947 | -0.00044736 |
| -4 | 0.00048248 | -0.00192993 |
| -3 | 0.00207909 | -0.00623728 |
| -2 | 0.04180923 | -0.08361847 |
| -1 | 0.40171191 | -0.40171191 |
| -0.5 | 0.04470705 | -0.02235353 |
| 0 | 0.08483290 | 0.00000000 |
| 1 | 0.31697909 | 0.31697909 |
| 1.5 | 0.04529632 | 0.06794448 |
| 2 | 0.05844299 | 0.11688598 |
| 3 | 0.00259645 | 0.00778935 |
| 4 | 0.00076323 | 0.00305292 |
| 5 | 0.00014491 | 0.00072453 |
| 6 | 0.00003774 | 0.00022646 |
| 7 | 0.00000609 | 0.00004263 |
| 8 | 0.00000066 | 0.00000526 |
| Total | 1.00000000 | -0.00277282 |
The table above reflects the following:
- House edge = 0.28%
- Variance = 1.303
- Standard deviation = 1.142
Probability of Net Win
I'm frequently asked about the probability of a net win in blackjack. The following table answers that question.
Summarized Net Win in Blackjack
The next three tables break down the possible events by whether the first action was to hit, stand, or surrender; double; or split.
Net Win when Hitting, Standing, or Surrendering First Action
| Event | Total | Probability | Return |
|---|---|---|---|
| 1.5 | 77147473 | 0.05144768 | 0.07717152 |
| 1 | 537410636 | 0.35838544 | 0.35838544 |
| 0 | 127597398 | 0.08509145 | 0 |
| -0.5 | 76163623 | 0.05079158 | -0.02539579 |
| -1 | 681213441 | 0.45428386 | -0.45428386 |
| Total | 1499532571 | 1 | -0.04412269 |
Net Win when Doubling First Action
| Event | Total | Probability | Return |
|---|---|---|---|
| 2 | 89463603 | 0.54980265 | 1.09960529 |
| 0 | 11301274 | 0.06945249 | 0 |
| -2 | 61954607 | 0.38074486 | -0.76148972 |
| Total | 162719484 | 1 | 0.33811558 |
Net Win when Splitting First Action
| Event | Total | Probability | Return |
|---|---|---|---|
| 8 | 1079 | 0.00002554 | 0.00020428 |
| 7 | 10440 | 0.00024707 | 0.00172948 |
| 6 | 64099 | 0.00151694 | 0.00910166 |
| 5 | 247638 | 0.00586051 | 0.02930255 |
| 4 | 1307719 | 0.030948 | 0.123792 |
| 3 | 4437365 | 0.10501306 | 0.31503917 |
| 2 | 10222578 | 0.24192379 | 0.48384758 |
| 1 | 2822458 | 0.06679526 | 0.06679526 |
| 0 | 5621675 | 0.1330405 | 0 |
| -1 | 3520209 | 0.08330798 | -0.08330798 |
| -2 | 9425393 | 0.2230579 | -0.4461158 |
| -3 | 3559202 | 0.08423077 | -0.25269231 |
| -4 | 828010 | 0.01959538 | -0.07838153 |
| -5 | 152687 | 0.00361343 | -0.01806717 |
| -6 | 30536 | 0.00072265 | -0.00433592 |
| -7 | 3972 | 0.000094 | -0.000658 |
| -8 | 305 | 0.00000722 | -0.00005774 |
| Total | 42255365 | 1 | 0.14619552 |
Liberal Strip Rules — Playing Two Hands at a Time
The following table shows the net result playing two hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the two hands.
6 Decks S17 DA2 DAS LS RSA P3X — Two Hands
| Net win | Probability | Return |
|---|---|---|
| -14 | 0.00000000 | 0.00000000 |
| -13 | 0.00000000 | -0.00000001 |
| -12 | 0.00000001 | -0.00000006 |
| -11 | 0.00000003 | -0.00000035 |
| -10 | 0.00000023 | -0.00000228 |
| -9 | 0.00000163 | -0.00001464 |
| -8 | 0.00001040 | -0.00008324 |
| -7.5 | 0.00000000 | -0.00000003 |
| -7 | 0.00005327 | -0.00037288 |
| -6.5 | 0.00000009 | -0.00000061 |
| -6 | 0.00024527 | -0.00147159 |
| -5.5 | 0.00000114 | -0.00000629 |
| -5 | 0.00106847 | -0.00534234 |
| -4.5 | 0.00000967 | -0.00004352 |
| -4 | 0.00654661 | -0.02618644 |
| -3.5 | 0.00005733 | -0.00020065 |
| -3 | 0.04607814 | -0.13823442 |
| -2.5 | 0.00214887 | -0.00537218 |
| -2 | 0.23285866 | -0.46571732 |
| -1.5 | 0.03547663 | -0.05321495 |
| -1 | 0.09903321 | -0.09903321 |
| -0.5 | 0.01386072 | -0.00693036 |
| 0 | 0.14677504 | 0.00000000 |
| 0.5 | 0.05888290 | 0.02944145 |
| 1 | 0.06026238 | 0.06026238 |
| 1.5 | 0.01030563 | 0.01545845 |
| 2 | 0.17250085 | 0.34500170 |
| 2.5 | 0.03020186 | 0.07550465 |
| 3 | 0.06443204 | 0.19329612 |
| 3.5 | 0.00559850 | 0.01959474 |
| 4 | 0.01072401 | 0.04289604 |
| 4.5 | 0.00024927 | 0.00112171 |
| 5 | 0.00187139 | 0.00935695 |
| 5.5 | 0.00007341 | 0.00040373 |
| 6 | 0.00049405 | 0.00296428 |
| 6.5 | 0.00001414 | 0.00009193 |
| 7 | 0.00012404 | 0.00086825 |
| 7.5 | 0.00000369 | 0.00002767 |
| 8 | 0.00002933 | 0.00023466 |
| 8.5 | 0.00000060 | 0.00000508 |
| 9 | 0.00000543 | 0.00004888 |
| 9.5 | 0.00000007 | 0.00000063 |
| 10 | 0.00000083 | 0.00000834 |
| 11 | 0.00000013 | 0.00000141 |
| 12 | 0.00000002 | 0.00000028 |
| 13 | 0.00000000 | 0.00000005 |
| 14 | 0.00000000 | 0.00000001 |
| Total | 1.00000000 | -0.00563798 |
The table above reflects the following:
- House edge = 0.28%
- Variance per round = 3.565
- Variance per hand = 1.782
- Standard deviation per hand= 1.335
Liberal Strip Rules — Playing Three Hands at a Time
The following table shows the net result playing three hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the three hands.
6 Decks S17 DA2 DAS LS RSA P3X — Three Hands
| Net win | Probability | Return |
|---|---|---|
| -16 | 0.00000000 | -0.00000001 |
| -15 | 0.00000000 | -0.00000001 |
| -14 | 0.00000001 | -0.00000007 |
| -13 | 0.00000003 | -0.00000041 |
| -12 | 0.00000018 | -0.00000218 |
| -11 | 0.00000100 | -0.00001099 |
| -10.5 | 0.00000000 | 0.00000000 |
| -10 | 0.00000531 | -0.00005309 |
| -9.5 | 0.00000001 | -0.00000006 |
| -9 | 0.00002581 | -0.00023228 |
| -8.5 | 0.00000005 | -0.00000047 |
| -8 | 0.00011292 | -0.00090339 |
| -7.5 | 0.00000049 | -0.00000370 |
| -7 | 0.00046097 | -0.00322680 |
| -6.5 | 0.00000397 | -0.00002581 |
| -6 | 0.00197390 | -0.01184341 |
| -5.5 | 0.00002622 | -0.00014419 |
| -5 | 0.00969361 | -0.04846807 |
| -4.5 | 0.00022638 | -0.00101870 |
| -4 | 0.04183392 | -0.16733566 |
| -3.5 | 0.00319799 | -0.01119297 |
| -3 | 0.15826947 | -0.47480842 |
| -2.5 | 0.02641456 | -0.06603640 |
| -2 | 0.08893658 | -0.17787317 |
| -1.5 | 0.02183548 | -0.03275322 |
| -1 | 0.09681697 | -0.09681697 |
| -0.5 | 0.04992545 | -0.02496273 |
| 0 | 0.06712076 | 0.00000000 |
| 0.5 | 0.02111145 | 0.01055572 |
| 1 | 0.08978272 | 0.08978272 |
| 1.5 | 0.03789943 | 0.05684914 |
| 2 | 0.04349592 | 0.08699183 |
| 2.5 | 0.01123447 | 0.02808618 |
| 3 | 0.10813504 | 0.32440511 |
| 3.5 | 0.02489093 | 0.08711825 |
| 4 | 0.06196736 | 0.24786943 |
| 4.5 | 0.00906613 | 0.04079759 |
| 5 | 0.01805409 | 0.09027044 |
| 5.5 | 0.00154269 | 0.00848480 |
| 6 | 0.00409323 | 0.02455940 |
| 6.5 | 0.00027059 | 0.00175885 |
| 7 | 0.00107315 | 0.00751203 |
| 7.5 | 0.00007208 | 0.00054062 |
| 8 | 0.00030105 | 0.00240840 |
| 8.5 | 0.00001824 | 0.00015505 |
| 9 | 0.00008014 | 0.00072126 |
| 9.5 | 0.00000431 | 0.00004096 |
| 10 | 0.00001901 | 0.00019010 |
| 10.5 | 0.00000081 | 0.00000846 |
| 11 | 0.00000398 | 0.00004379 |
| 11.5 | 0.00000013 | 0.00000144 |
| 12 | 0.00000078 | 0.00000939 |
| 12.5 | 0.00000002 | 0.00000023 |
| 13 | 0.00000016 | 0.00000214 |
| 13.5 | 0.00000001 | 0.00000008 |
| 14 | 0.00000003 | 0.00000045 |
| 14.5 | 0.00000000 | 0.00000001 |
| 15 | 0.00000001 | 0.00000009 |
| 15.5 | 0.00000000 | 0.00000000 |
| 16 | 0.00000000 | 0.00000002 |
| 17 | 0.00000000 | 0.00000001 |
| Total | 1.00000000 | -0.00854917 |
The table above reflects the following:
- House edge = 0.285%
- Variance per round = 6.785
- Variance per hand = 2.262
- Standard deviation per hand= 1.504
