Blackjack Apprenticeship Risk Of Ruin

Part of the full version of SBA is the Explanation of Blackjack Statisticsdocument. This electronic book explains in detail the important notions in blackjack: advantage, standard deviation, how to calculate the optimal bets, risk-of-ruin, the importance of risk-averse strategy indices, and more.

Currently, much of the advantage play community focuses on the risk of losing a bank before doubling it without resizing after session wins or losses. Little has been written on long term money management. This article focuses on managing long term risk while keeping the method of doubling bankrolls.

Currently, there are two tools available to the advantage player for long term risk management. The optimal tool is the Kelly criterion. But to use Kelly's technology properly, each bet should be sized according to the player's current bank. The more one does not resize one's bet sizes, the less one is playing by the mathematics of the Kelly criterion. We shall show that never resizing after setting initial betting levels is disastrous. The second tool is Don Schlesinger's trip risk of ruin formula, which does not take into account either the current gambling paradigm of doubling a bank or the problem of resizing bets.

  1. Risk of Ruin Calculator Introduction. The purpose of this calculator is to estimate the probability of ruin, given a positive expected value, standard deviation, bankroll, and infinite play. The calculator assumes the player flat betting and the odds of every trial are the same.
  2. CVCX Online Calculators - Risk Risk of Ruin Given No Goal but a Time Constraint (Trip Ruin) This is the trip ruin formula in Blackjack Attack page 132. Risk is still calculated, but for a specified number of hands. The risk is lower than with the calculator with no time constraint since you have a quit point.
  3. If you have that kind of bankroll, your risk of ruin is just 1%. On the other hand, if your tolerance for risk is better than that, you could get away with a much smaller bankroll—maybe 200 units. You’d still need $20,000, but you’d be able to play at that level. Your risk of ruin goes way up, though—to 40%.

I. No Reinvestment

As an illustration of the above concept, suppose that a gambler is currently making his living at blackjack. He has a $20,000 stake which he can double twice a year. He uses the $40,000 as living expenses. Unfortunately, even with conservative play he is destined to have a limited career. While it may be very liberal, say he is playing via half the Kelly criterion without resizing. Then the probability of doubling a bank before going bust (RoR) is about 1.8%. The probability of going bust before doubling a bank n times is equal to

(1) CRoR=1-(1-RoR)^n.

Here, CRoR stands for cumulative risk of ruin. Unfortunately, this means that that the probability of not surviving 5 years in the above scenario is (1-13%)^10=17%. The probability of not surviving 10 years is (1-13%)^20=31.1%. If RoR is greater than zero, then the above formula approaches zero as n grows large. The long run is a bad thing in this scenario.

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II. Constant Reinvestment

Certainly there is some happy medium between spending all casino winnings and not spending anything that still provides an acceptable long term risk of ruin. The remainder of this article will explore the risk involved with various reinvestment strategies. If a gambler plays with a decreasing risk of ruin, he might be around indefinitely. If r_k is the probability of not doubling our kth bank, (1) may be replaced with

where k runs from one until the end of one's advantage playing career. It can be shown that if r_k shrinks sufficiently swiftly, then (2) will not approach one and failure is not certain in the long run. However, it must be stressed that just any decreasing sequence of r_k's will do.

The main tool in risk of ruin calculations is certainly

(3) r~ exp(-2BE/s ^2),

where r is the risk of ruin, B is a bankroll, E is expected value per hand, and s is the standard deviation per hand. Versions of this equation have been presented in Blackjack Forum, Blackjack Attack, and the International Conference on Gambling. If a game with fixed standard deviation and expected value is played, then one may use (3) to express (2) purely as a function of bankroll. It follows that

(4) CRoR= 1-? (1-exp(-2B_k E/s ^2)).

=1-? (1-r_k)

where E and s are constants, and B_k is the size in units of the kth bank. For example, say one is playing a double deck game with E=0.02 and s =2.5. Let the initial bank be 500 units. Say 200 units are reinvested in our bank whenever it is doubled and a gambling career is the time to double 7 banks. Then r_k=exp(-(B+(k-1)200)E/s ^2), and by using a calculator it can be determined that


In fact, if the number of banks is increased to a number greater than 7, then CRoR will remain 5.58% accurate to 0.1%. So it has been shown that it is possible to be a professional gambler for an indefinite length of time.

Unfortunately, the problem of increasing our unit size has not been addressed. In the above example the size of the bank in increasing, but a gambler following the plan will play with a constant expectation in dollar terms. If someone is satisfied with their current expected win level or may find it difficult to increase their unit because of practical terms, then this may be a satisfactory model of risk.

III. Proportional Reinvestment

Instead of reinvesting a constant number of units for every bank doubled, suppose one reinvests a fixed fraction of winnings back into a bankroll after doubling a bank. For example, consider the case where after doubling a bank, 20% of all winnings are put back into the next bank. If s and E remain at 2.5 and 0.02, and B=500 is the initial bank size, then the kth bank is equal to B(1+0.2)^(k-1)=B(1.2)^(k-1) units. It follows that the risk of ruin for the kth bank is equal to r_k=exp(-2(1.2)^(k-1)BE/s ^2) and the cumulative risk of ruin after 5 banks is equal to


In fact, if the number of banks is increased to any number greater than 5, CRoR remains at 7.6% accurate to 0.1%.

The problem of unit resizing can now be addressed. Let s , E, and B be as above. Say that when a bank is doubled, 40% of all winnings are put back into the next bank. Also, when a bank is doubled one can increase the size of the units by 20%. Then the kth bank is of size (B(1+0.4)^(k-1))/(1+0.2)^(k-1). In general, if a is the fraction of winnings reinvested in the bank and b is the percentage the unit size is increased, then the kth bank is equal to

(5) B((1+a)/(1+b))^(k-1)

And the risk of ruin on the kth bank is equal to

(6) r_k=exp(-2((1+a)/(1+b))^(k-1)*BE/s ^2).

It follows that the cumulative risk of ruin is equal to

(7) CRoR=1-? (1- exp(-2((1+a)/(1+b))^(k-1)*BE/s ^2)),

Where k goes from 1 to the end of one's gambling career. Note that (7) approaches one if a<=b. So the increase in unit size must be less than the proportion reinvested into the bankroll.

If the preceding variables are used for s, E, B, a, and b, then CRoR is equal to 6.94% for all k greater than 3. Note that in this scenario, one's expected value in dollars is increasing 20% after doubling each bank. So for example, if one is making $20 per hour on an initial bankroll, then after only 5 banks one's expected value is $20*(1.2)^5=$50 per hour. After 10 banks, one's expected value is $124 per hour. Note that 40% of all winnings are reinvested though, so one is only making 60% of the above figures for living expenses. Also, one may also pay taxes on all of one's winnings, not just the portion used for living expenses.

  1. Managing Risk and Different Playing Styles

The previous method may not be how one wants to manage a bankroll, but gives the tools necessary for measuring the total amount of long term risk in one's gambling career. For example, one may note that having an arbitrarily large expected value in the long run is unrealistic. Say if one determines that the maximal hourly expected value cannot be greater than say $500. Then after E reaches $500, simply set b=0.

Another adjustment that may be made is to fix one's long term risk of ruin by picking a and b after each bank is doubled The first thing to realize in this regard is that we are only concerned with the ratio (1+a)/(1+b). Then use Excel or some other tool, solve for the value of (1+a)/(1+b) that sets CRoR to the desired level if reasonable values are used, CRoR should converge within almost any desired accuracy within 20 banks.

Some blackjack players are so preoccupied with mastering perfect basic strategy and card counting that they neglect their money management. In blackjack, just like in any other casino-banked game, managing one’s bankroll adequately is of great significance.

Having said that, we would also like to point out bankroll management is powerless when it comes to decreasing the house edge. What it does help with is longevity, or preserving your blackjack bankroll for a longer period of time. No matter how perfect your play is, you are guaranteed to lose your money without discipline and proper bankroll management.

Building a Bankroll – How Much Money Do You Need to Play Blackjack?

Let’s start by specifying that your bankroll is the money you have set aside strictly for the purpose of playing blackjack. We suspect you already know this but just to play it safe, we shall say it again – you should never use money you need to cover your day-to-day expenses for playing blackjack, regardless of your level of skill or previous experience.

Our advice is to place your blackjack bankroll in a separate account and withdraw from it when you plan to attack the blackjack tables. Once you finish with the assault, you go back and deposit whatever you have left alongside any winnings you have generated during the session.

You should leave your bankroll alone in the beginning and avoid using it for any non-blackjack-related purchases. Once you succeed in building your bankroll, you can reward yourself by buying something with some of the winnings you have generated.

Table Limits and Session Bankrolls

With this clarification out of the way, we warn you there is no uniform bankroll size that applies to absolutely all blackjack players. The edge skilled players get inevitably manifests itself over the long term. Anything can happen over the course of a single session, a week, or even a few months.

Experiencing short-term losses, even if you are an accurate card counter, is hardly anything unheard of. The bottom line is as a serious blackjack player, you need a bankroll that is large enough to withstand the losses you may incur on a short timescale.

The overall amount you allocate for blackjack play should be broken down into smaller session bankrolls. How much you allocate for a single session is closely linked to what table limits you play.

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If there are lots of casinos in your area but you have limited funds for blackjack play at your disposal, the smartest thing to do is scout the different gambling halls and find a table with low enough limits to accommodate your small bankroll. Provided that there is a single casino with high limits in your city, you better wait until you save a sufficiently large bankroll to play such stakes.

Show MoreHide MoreA session bankroll should be at least 50 times the lowest bet at the table. This is the bare minimum, recommended for basic strategy players and flat bettors. Respectively, players who count cards and move their bets with the true count are recommended to put aside at least 100 times their top bets.

Thus, if there are $10 tables in your vicinity and you flat bet at this minimum with basic strategy, your session’s bankroll should be at least $500. Your max bet should not exceed the amount of $10 under any circumstances. Provided that you are a novice card counter who uses a less aggressive bet spread like 1 to 5, you will need a session bankroll of at least $5,000.

Each number 1 through 5 corresponds to the number of base bets you need to wager when you move with the true count. You put out 5 units or $50 on a count of +5 or higher, 4 units or $40 on a count of +4, and so on. One unit of $10 is wagered on a count of +1 as well as on neutral and negative counts.

Blackjack Apprenticeship Risk Of Ruin The Following

Evaluating Your Risk of Ruin

Disciplined players who exercise good money management are well-acquainted with the term “Risk of Ruin”, abbreviated as RoR. For those of you who are not, RoR denotes the probability of a given player losing their entire bankroll.

There are several values you need to take into account when estimating your Risk of Ruin, including your standard deviation, your bankroll in units, and your win rate per every hundred hands. There are free RoR calculators on the web players can use to accurately estimate the likelihood of busting their full bankrolls. Your other option is to use blackjack simulators that can calculate the RoR for you.

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We can distinguish between two types of Risk of Ruin, namely session RoR and the RoR for players’ full bankroll. The former denotes the likelihood of the player losing their entire bankroll for the session while the latter shows you the probability of busting your overall lifetime bankroll.

To give you an example, let’s suppose you have a session bankroll of $2,000, play perfect basic strategy, and flat bet $10 per hand. You have 200 base betting units at your disposal. The software you are using has calculated that you have a session RoR of 18%.

This means that eventually you will end up losing your $2,000 around 18% of the time. And the opposite, your bankroll will increase 82% of the time. Meanwhile, if you cut your bankroll in half to $1,000, or 100 units, your RoR will jump to nearly 32%, which exceeds the tolerable limits. In the other 68% of the time, you will increase the bankroll.

It is important to specify that different players are willing to put up with different RoR percentages. At the end of the day, this is all a matter of individual tolerance. The bottom line is the bigger your bankroll is and the more base betting units you have, the lower your RoR will be.

Understanding Standard Deviation

The term standard deviation (SD) is normally used in mathematical statistics in relation to the distribution of expected results. In blackjack, it denotes the distribution of players’ results within a range of probable outcomes.

It tells you how frequently a specific outcome will deviate from your expected average. This is important because it enables you to assess whether you are playing a losing or a winning game as well as to decide how big your bankroll should be for any given session.

It is unrealistic to think you can win each and every blackjack session, even if you are perfect at basic strategy and count cards with great accuracy. A low standard deviation indicates the actual results fall closely within one’s expectations.

We shall explain how standard deviation works with a simple coin-flipping example. A coin has a 50% chance of landing on tails and a 50% chance of landing on heads. Yet, you cannot expect the coin to land precisely 50 times on tails and 50 times on heads in every 100 trials, or at least not in the short term. Sometimes it may land only 45 times on tails and 55 times on heads which happens roughly 2/3 of the time or around 68.3%.

Show MoreHide MoreIn the context of blackjack, the standard deviation of a single hand you play in a six-deck game is estimated at 1.14. Thus, you are expected to win or lose roughly 1.14 bet units around 68.3% of the time within one standard deviation, 2.28 betting units will be lost/won 95% of the time within two standard deviations and 3.42 units will be lost/won 99.7% of the time within three standard deviations. The distribution of these results is shown on the so-called Gaussian bell-curves.

Knowing their standard deviation enables players to calculate the probability of winning or losing a given number of units over the course of a certain number of hands. You do this by multiplying your standard deviation by the square root of the number of hands you play.

So if your sample size involves 400 hands with a standard deviation of 1.14, you can expect to lose or win √400 x 1.14 = 20 x 1.14 = 22.8 betting units around 68.3% of the time. Respectively, 95% of the time, you can expect results within two standard deviations where you will lose √400 x 2.28 = 20 x 2.28 = 45.6 betting units over the course of 400 hands. And finally within three standard deviations, you will lose √400 x 3.42 = 20 x 3.42 = 68.4 betting units every 400 hands 99.7% of the time.

Standard deviation may be complex to understand if you are a novice but is nevertheless of great importance. You need it when calculating your RoR, which in turn helps you determine the bankroll you need. Do not be intimidated, however, as you can figure out what your RoR is by using a simulator software or one of the online RoR calculators.

House Edge and Hourly Losses

The beauty of using basic strategy is that it reduces the house edge in blackjack to such an extent that you are nearly playing a break-even game. Yet, basic strategy is not powerful enough to completely overcome the built-in casino advantage.

Even if you are perfect at basic strategy, the house edge will inevitably cause a dent in your blackjack bankroll over the long run. This dent, however, will be far more significant if you rely on gut feelings and hunches instead of using the optimal strategy.

Blackjack Apprenticeship Risk Of Ruin Credit

Knowing the house edge of a blackjack game helps you calculate the hourly losses you can expect to incur in the long term. Suppose you choose a table with more liberal rules like those offered across Las Vegas Strip casinos where the house edge revolves around 0.36%.

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You multiply this percentage by the average number of hands you play per hour and your average bet size. Assuming you are a recreational player who joins mostly full tables and bets $30 per hand on average, you will be able to go through roughly 80 hands per hour.

Therefore, the long-term hourly losses you can expect to see will amount to ($30 x 80 hands x 0.36)/100 = 864/100 = $8.64.You will inevitably arrive at this figure when you get enough playing hours under your belt. By “enough”, we mean tens of thousands of hours as anything can happen in the short run.

Unlike basic strategy players who are practically betting on a negative EV game, skilled and disciplined card counters are able to overcome the house edge. They have an advantage of around 1% at six-deck games with decent rules.

This enables them to grow their bankrolls overtime instead of incurring long-term losses. They calculate their expected hourly winnings with the same formula, i.e. by multiplying their edge by the average bet size and the number of hands they play per hour.

Respectively, an accurate counter who plays heads-up at an empty table with at a 1% advantage and goes through 100 bets of $30 per hour can expect long-term hourly returns of ($30 x 100 hands x 1)/100 = 3,000 / 100 = $30.

Handling Losing Sessions

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Variance is inherent to all casino games, including blackjack. All players, no matter how skilled they are, will inevitably end up going through some losing sessions. Knowing how to handle these and when to call it quits is of great significance for preserving your bankroll.

Needless to say, chasing your losses is a terrible idea. The rule of thumb all smart blackjack players should follow is to always leave a table before they have busted their entire session bankroll. The general recommendation is to throw in the towel when you are left with fewer than six betting units.

So if you bet $50 per hand, you must ensure you have at least $300 before you continue playing; if you wager $100, you end the session when you are down to less than $600 and so on.

The reason for this is simple – you need enough money to back up any potential splitting and doubling decisions in line with basic strategy. The bottom line is you should never stay at the table if you are so underbanked that you can no longer exercise the optimal playing decisions. Doing the opposite will ultimately cost you money in the long run.