A classic blackjack strategy that tells the player how to act combined with a good betting system can lead to big winnings in the long run. Here is a basic blackjack strategy that will help you take the optimal decision in each possible situation.

• When you read about the expected house edge in blackjack games based on the rules the percentages are based on perfect basic strategy. If you don’t use the proper strategy you give the house a higher edge against you. This can add an extra one or two percent to the house edge, depending on how far from the proper strategy you stray.
• Oscar’s System – Blackjack Betting Strategies Oscar’s System was proposed in the 1960’s in a book by mathematician Dr. It is a fairly decent system. The goal of the strategy is to win exactly one unit per series.

My last post was about some common mistakes when bettingor gambling, even with a basic understanding of probability. This post is going totalk about the other side: optimal betting strategies using some veryinteresting results from some very famous mathematicians in the 50s and 60s.I'll spend a bit of time introducing some new concepts (at least to me), setting up theproblem and digging into some of the math. We'll be looking at it from thelens of our simplest probability problem: the coin flip. A note: I will not becovering the part that shows you how to make a fortune -- that's an exercisebest left to the reader.

Background

History

There is an incredibly fascinating history surrounding themathematics of gambling and optimal betting strategies. The optimalbetting strategy, more commonly known as the Kelly Criterion, was developed in the 50s byJ. L. Kelly , a scientistworking at Bell Labs on data compression schemes at the time. In 1956, he madean ingenious connection between his colleague's (Shannon) work on information theory,gambling, and a television game show publishing his new findings in a papertitled A New Interpretation of Information Rate (whose original title wasInformation Theory and Gambling).

The paper remained unnoticed until the 1960s when an MIT student named Ed Thorptold Shannon about his card-counting scheme to beat blackjack. Kelly's paperwas referred to him, and Thorp started using it to amass a small fortune usingKelly's optimal betting strategy along with his card-counting system. Thorpand his colleagues later went on to use the Kelly Criterion in othervaried gambling applications such as horse racing, sports betting, and even thestock market. Thorp's hedge fund outperformed many of his peers and it wasthis success that made Wall Street take notice of the Kelly Criterion. There is agreat book called Fortune's Formula 1 that details the stories andadventures surrounding these brilliant minds.

Surely, Almost Surely

In probability theory, there are two terms that distinguish very similarconditions: 'sure' and 'almost sure'.If an event is sure, then it always happens. That is, it is not possible forany other outcome to occur. If an event is almost sure then it occurs withprobability 1. That is, theoretically there might be an outcome not belonging tothis event that can occur, but the probability is so small that it's smallerthan any fixed positive probability, and therefore must be 0. This is kind ofabstract, so let's take a look at an example (from Wikipedia).

Imagine we have a unit square where we're randomly throwing point-sized darts thatwill land inside the square with a uniform distribution. For the entire square(light blue), it's easy to see that it makes up the entire sample space, so we wouldsay that the dart will surely land within the unit square because there is noother possible outcome.

Further, the probability of landing in any given region is the ratio of itsarea to the ratio of the total unit square, simplifying to just the area of agiven region. For example, taking the top left corner (dark blue), whichis 0.5 units x 0.5 units, we could conclude that (P(text{dart lands indark blue region}) = (0.5)(0.5) = 0.25).

Now here's the interesting part, notice that there is a small red dot in theupper left corner. Imagine this is just a single point at the upper leftcorner on this unit square. What is the probability that the dart lands on thered dot? Since the red dot has an area of (0), (P(text{dart landson red dot}) = 0). So we could say that the dart almost surely does not landon the red dot. That is, theoretically it could, but the probability of doingso is (0). The same argument can be made for every point in the square.

The dart actually does land on a single point of the square though, so eventhough the probability of landing on that point is (0), it still doesoccur. For these situations, it's not sure that we won't hit that specificpoint but it's almost sure. A subtle difference but quite important one.

Optimal Betting 2

Optimal Betting with Coin Tossing

Imagine playing a game with an infinite wealthy opponent who will always takean even bet made on repeated independent tosses of a biased coin.Further, let the probability of winning be (p > frac{1}{2}) andlosing be (q = 1 - p)3, so we have a positive overall expected valuefor the game 4. You start with (X_0) of initialcapital. Question: How much should we bet each time?

This can be made a bit more concrete by putting some numbers to it.Let's say our coin lands on heads with a chance of (p=0.53),which means tails must be (q=1-p=0.47). Our initial bankroll is(X_0=$100,000). How much of this ($100,000) should we bet on the firstplay?

Let's formalize the problem using some mathematics. Denote our remaining capitalafter the k'th toss as (X_k) and on the k'th toss we can bet (0leq B_k leq X_{k-1}). Let's use a variable (T_k = 1) if thek'th trial is a win, and (T_k=-1) for a loss. Then for the n'th toss, we have:

One possible strategy we could use is to maximize the expected value of(X_n). Let's take a look at that:

Since (p - q > 0) this will have a positive expected payoff. To maximize(E(X_n)), we should maximize (E(B_k)) (this is the only variable wecan play with), which translates to betting our entire bankroll at each toss.For example, on the first toss bet (B_0 = X_0), on the second toss (if we wonthe first one) bet (B_1 = 2X_0) and so on. It doesn't take amathematician to know that is not a good strategy. Why? The probability ofruin is almost sure (ruin occurs when (X_k = 0) on the k'th toss).

If we're betting our entire bankroll, then we only need one loss to lose allour money. The probability of ruin is then (1 - p^n) for (n) tosses (everyoutcome except winning on every toss). Taking the limit as (n) approaches infinity:

So we can see that this aggressive strategy is almost surely 5 going to result in ruin.

Another strategy might be to try and minimize ruin. You can probably already intuitthat this strategy involves making the minimum bet. From Equation 2, this isnot desirable because it will also minimize our expected return. This suggests that wewant a strategy that is in between the minimum bet and betting everything (duh!).The result is the Kelly Criterion.

The Kelly Criterion

Since our maximum bet is limited by our current bankroll, it seems plausible thatthe optimal strategy will always bet relative to our current bankroll. Tosimplify the math, we assume that the money is infinitely divisible. However,it should be noted that this limitation doesn't really matter too much when ourcapital is relatively large compared to the minimum divisible unit (thinkmillions vs. cents).

If on every toss, we bet a fraction of our bankroll (known as 'fixed fraction'betting), (B_k = fX_{k-1}), where (0 leq f leq 1), we canderive an equation for our bankroll after (S) successes and (F) failuresin (S+F=n) trials:

Notice that we can't technically ever get to (0) but practically there is a minimumbet and if we go below it, we are basically ruined. We can just re-interpretruin in this manner. That is, ruin for a certain strategy is when we willalmost surely go below some small positive integer (epsilon) as thenumber of trials (n) grows i.e., (lim_{nrightarrow infty}P(X_nleq epsilon) = 1).

Now let's setup what we're trying to maximize.We saw that trying to maximize the expected return leads us to almost surelyruin. Instead, Kelly chose to maximize the expected exponential growth rate.Let's see what that means by first looking at the ratio of current bankroll toour starting bankroll:

So (G(f)) represents the exponent (base (e)) on how fast ourbankroll is growing. Substituting Equation 4 into (G(f)):

Now since (G(f)) is a random variable, we want to maximize the expectedvalue of it (which we denote as (g(f))):

The last line simplifies because the expected proportion of successes andfailures is just their probabilities 6. Now all we have to do is a simpleexercise in calculus to find the optimal value (f^*) that maximizes (g(f)):

Blackjack Betting Strategy 1 3 2 6

So we now have our optimal betting criterion (for even bets), fractional betswith (f^*=p-q).

Another interesting behavior of varying our fractional bets can be gleaned bygraphing (G(f))7:

We can see that our (f^*) maximizes the growth rate. However, there is a point(f_c) where our growth rate becomes negative. This implies that if weover-bet (f > f_c), we will almost surely reach ruin (because we have anegative growth rate). The following (summarized) theorem from Thorp's paperstates this more precisely:

1. If (g(f) > 0), then (lim_{nrightarrow infty}X_n = infty) almost surely.

2. If (g(f) < 0), then (lim_{nrightarrow infty}X_n = 0) almost surely.

3. Given a strategy (Theta^*) and any other 'essentially different strategy' (Theta), we have (lim_{nrightarrow infty}frac{X_n(Theta^*)}{X_n(Theta)} = infty) almost surely.

From this theorem, we can see that if we pick a fraction such that (g(f)> 0), then we'll almost surely tend towards an increasing bankroll.Conversely, if we pick a fraction (g(f)<0), then we will almost surelyresult in ruin. This matches up with our intuition that over-betting iscounter-productive.

(Continued from Example 1)Suppose we have our even-bet coin toss game and the probability of heads is(p=0.53) and probability of tails is (q=0.47). Our initialbankroll is ($100,000) (big enough that the minimum bet isn't reallysignificant). Applying our optimal betting criteria, on our first playwe should bet (f=p-q=0.53-0.47=0.06) or (6%) of our bankroll, translating to($100,000 * 6% = $6,000). Assuming we win the first play, we should bet($106,000 * 6% = $6,360) and so on.

If we bet less than (6%), we will still be increasing our bankroll but not atthe optimal rate. We can also bet more than (6%) up to the theoretical point (f_c)such that (g(f_c)=0) with the same result.We can numerically determine this turning point, which in this case is(f_c approx 0.11973). So betting more than roughly 11.9% will almostsurely cause us ruin.

We can also compute the expected exponential growth rate using our optimal(f^*= 0.06):

So after (n) plays, a player can expect his bankroll to be(e^{0.001801n}) times larger. A doubling time can be computedby setting (e^{0.001801n}=2), resulting in (napprox 385) plays.

Betting with Uneven Payoffs and Other Variations

We've so far only looked at games with even payoffs. We can generalize this result.If for each unit wagered, you can win (b) units, we can derive a modified versionof Equation 7:

Solving for the optimum yields (f^*=frac{bp-q}{b}).

Another variation is when you can make multiple simultaneous bets such as whenmultiple players share a single bankroll. Going through a similar exercise, wecan derive values for (f_1^*, f_2^*, ldots) assuming the games playedare independent. When two players are playing the same game (e.g. same tablefor Blackjack), the bets are correlated and adjustments must be made.Additionally, we can analyze more complex situations such as continuous (ornearly continuous) outcomes like the stock market which require a more thoroughanalysis using more complex math. See Thorp's paper for more details.

Conclusion

Kelly's optimal betting criterion is an incredibly interesting mathematicalresult. However, perhaps what is more interesting is that this theoretical resultwas put into practice by some of the very mathematicians that worked on it!Thorp has had wild success applying it in various situations such assports betting, Blackjack and the stock market. Of course by itself thecriterion isn't much use, it is only once you've found a game that has apositive expected value that you can put it to use. I would go into how to dothat but I think I've written enough for one day and as I said, it's best leftas an exercise to the reader.

References and Further Reading

• The Kelly Criterion in Blackjack Sports Betting, and the Stock Market by Edward O. Thorp.

• Optimal Gambling Systems for Favorable Games, E. O. Thorp, Review of the International Statistical Institute Vol. 37, No. 3 (1969), pp. 273-293 .

• William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. 2005. ISBN 978-0809045990. See also a brief biography of Kelly on William Poundstone's web page.

William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. 2005. ISBN 978-0809045990. See also a brief biography of Kelly on William Poundstone's web page.

This whole section just basically summarizes (with a bit more step-by-step for the math) the paper 'The Kelly Criterion in Blackjack Sports Betting, and the Stock Market'. So if you're really interested, it's probably best to check it out directly.

It doesn't really matter if the bias is heads or tails. The point is that you get to pick the winning side!

The expected value of winning for bet (B) is (Bp-Bq = B(p-q) > 0) since (p > q).

Almost surely here because it's theoretically possible that you can keep winning forever but it's such a small possibility that it basically can't happen. This is analogous to the red dot in the unit square.

The expected value of a binomial distribution (e.g. coin tossing) is just (np). So (np/n = p).

Image from 'The Kelly Criterion in Blackjack Sports Betting, and the Stock Market'.

Table Of Contents

Are you looking for the best blackjack strategy to maximise your winning odds when you play online? You'll love these tips.

Finding a way to get higher chances to win at blackjack shouldn't be so hard.

The basic strategy is the best way to play blackjack — and when you combine it with some strategy tips on bankroll management and what to do after you win, you have really reached the maximum you can.

In other words, if you are looking for the best blackjack betting strategy to increase your chances to win, you'll love this guide.

In the parts that follow you will find:

• And lots more

We also put together a collection of the best bonuses to play online. This way, you can use those freebies to test the blackjack basic strategy and all the tips in this article.

Blackjack Betting Strategy Chart

Register a FREE account and claim your bonus to play online blackjack!

What this guide is NOT

• A collection of tips to win at blackjack
• A list of systems that promise to make blackjack profitable
• The illusion that you can beat the game

Blackjack is a gambling game where the casino has an advantage. It doesn't matter what approach you take or what blackjack strategy you follow. In the long run, the casino always wins.

Why Follow a Blackjack Betting Strategy?

Let's face it - we have all been there. We have all looked for the best and the most effective strategy to win at blackjack after we lost our first hands online.

With the right blackjack strategy it's possible to increase your winning chances, and that's because blackjack isn't a gambling game like roulette, video poker, or slots.

If you follow the basic betting strategy and you learn how to use the information you have at your disposal, you can reach optimal play giving yourself the best chances to win include in the game.

Professional players have worked hard to 'solve' the game of blackjack and find how to play each hand to get the best winning odds.

They analysed how the number of decks in play affects your chances to win and how you can make the best decision based on the cards you see on the table.

• That's what they call the 'blackjack basic strategy. A list of dos and don'ts where you find exactly how you are supposed to play each and every hand you are dealt.

This is the best strategy for blackjack.

The goal of the blackjack basic strategy isn't to help you win at blackjack every time you play. That's impossible. Instead, it is to help you maximize your winning chances and to minimize your losses.

That's also the goal of this guide. We don't list all the strategies for blackjack out there. Instead, we focus only on the best blackjack strategies that are known to work and should help you get better results when you play.

Starting from the tips and tricks included in our cheat-sheet.

READ ALSO: How to play online blackjack with friends.

What Is the Blackjack Basic Strategy?

Knowing how to manage your money is as important in blackjack as it is in all gambling games. A correct 'bankroll management' is essential when you play — regardless of the blackjack strategies you apply to your games.

Knowing how to bet (and how much) and how to select the right game when you play online blackjack for real money is extremely important.

Once you have those basics covered, you can move into the basic blackjack strategy and learn how you are supposed to invest your money once the dealer gives you the initial two cards.

As soon as the cards reach you and the dealer opens one of their cards, you have to make your first choice. In a game of blackjack, that means you need to pick one of these options:

• Stand
• Hit
• Double
• Split
• Surrender

Do you know why the basic blackjack strategy is also considered to be the best way to discover how to win at blackjack? Because it uses basic logic.

Would you hit if your first two cards gave you 20 points? Of course not.

Would you stand if your starting hand was worth five points? Not a chance.

While these choices are very simple on hands like the ones I use in my example, things get (a lot) more complicated with different cards on the table.

Let me give you a few examples.

• How do you play soft 17 in blackjack?
• When do you hit on a 16?
• What to do with 15 in blackjack?

This is where the basic blackjack strategy comes in to help you get the best chances to win.

This collection of tips shows you the optimal play for every blackjack hands, calculating your winning odds in combination with the dealer's open card.

The basic strategy to win at blackjack has nothing to do with concepts like card counting or any of those tricks some 'clever' players adopt to beat the house.

This is a simple-to-use guideline that shows you exactly how you are supposed to play every possible hand of blackjack. Use it to know when it is recommended to hit, to stand, or to do something else.

Key:
S = Stand
H = Hit
Dh = Double (if not allowed, then hit)
Ds = Double (if not allowed, then stand)
SP = Split
SU = Surrender (if not allowed, then hit)

A Blackjack Cheat Sheet

To make things even easier for you and help you follow the best betting strategy whenever you play blackjack, we created a quick cheat-sheet with the basic strategy explained in full.

Download the blackjack cheat-sheet (it's free) and print it or save it on your mobile phone to keep it always with you when you play.

The blackjack basic strategy cheat-sheet is 100% legal and you can use it both when you play online and at a brick-and-mortar casino.

A Conservative Blackjack Strategy

Don't play high-stakes games unless you feel ready for them and you have enough money to spend on online games.

If you are reading this blackjack strategy guide and you are a beginner, you should be afraid of high-stakes games.

When you spend too much money on your games, you risk getting the opposite of what you want. You play to have fun and to find out how to get more chances to win at blackjack - and you end broke and frustrated.

The best way to play blackjack is to start with what I call a conservative approach. Because finding the right way top handle betting in blackjack is key to long-term fun.

Like many blackjack strategies suggest, you should increase the stakes only when things go well and any potential losses are compensated by the money you have already won, not the one you might win if you get lucky.

In other words, raise your bet every time you win.

Many players believe wins and losses come in streaks. They think you might lose five or six times in a row — but when you win…you are not going to do it just once.

Obviously, that's not true.

Blackjack is a gambling game and there are no patterns for you to exploit or streaks that you can use to win more money every time you play.

There are, however, smart betting strategies like the basic blackjack strategy that can help you increase the chances of a victory.

With that in mind, a conservative blackjack strategy works like this:

• Start from the lowest bet (e.g. $1)
• Increase your bet by 1 unit only after you win
• Go back to the original bet after you lose

When you use this blackjack strategy, you need to go back to the original bet as soon as you lose one hand. This is the best way to be in control of your money and end a gaming session in the black.

All you need to do is to make the right choices (use the basic strategy) and raise your stakes little by little - but only when you win.

READ ALSO: Bets sites to play live dealer blackjack online.

An Aggressive Approach to Blackjack: the Martingale Strategy

The Martingale betting strategy sits at the opposite side of the spectrum. This is the perfect blackjack strategy for those players who:

• Play blackjack to win a lot of money
• Can afford to lose a lot of money
• Don't fear the swing of big losses

In other words, this is the perfect blackjack strategy for big spenders but isn't one that you should try if you are still a beginner who is looking for ways to learn how to play blackjack.

The Martingale blackjack strategy is expensive. This is a blackjack system that will help you win often - but only when I have a lot of money to invest on your games.

If you can't spend at least $500 on your games - the Martingale is not the best blackjack strategy for you.

But if you do — continue reading to discover how to use this betting strategy to increase your chances to win.

In short:

The Martingale strategy is the exact opposite of the conservative approach to betting in blackjack we have just seen.

When you follow this strategy, you need to double your bet every time you lose. Which, in case of a losing streak, can become very expensive very quickly.

The rationale of this audacious blackjack betting strategy is that you will need only one win to go back to square one and recoup all your losses in one hand.

The caveat is: your bankroll needs to be deep enough for you to play and play until that one win comes.

Again, this isn't a betting strategy for beginners. If you decide to try it, make sure your bankroll covers at least 10 consecutive losses from the hand #1.

1-3-2-6 Blackjack Betting System

The 1-3-2-6 blackjack betting strategy is not as exciting as the Martingale strategy, and that's a good thing. This betting strategy doesn't require a huge bankroll and is a lot more suitable for players looking for a way to have better chances to win at blackjack without investing a fortune in the game.

The numbers in the strategy's name indicate the way you should bet after you win a hand of blackjack.

You first winning is the trigger that starts the sequence while your first losing hand is the one that ends everything.

This blackjack betting strategy is built to get the most value out of winning streaks. If you are lucky enough to use it at the beginning of a four-hand winning streak, you are in for a treat.

READ ALSO: How to play blackjack for beginners.

How to Win More Often at Blackjack: A Few Important Tips

We all want to win at blackjack every time we play, but we also know that's impossible.

What is possible, however, is to maximise your chances to win and get better winning odds when you play blackjack online. All it takes a few simple tips.

Follow them carefully, as these blackjack tips are as important as the best blackjack betting strategy.

• Don't spend money on the 'insurance'. The insurance bet is a money-draining option created to confuse beginners. All the best blackjack strategies fail to mention is for one simple reason: no real blackjack player will ever spend their money on it.

• Think about the house edge. Like we said when we went through some of the game's best betting strategies, you can't expect to win at blackjack every time you play.

  Like all other casino games, the house edge is what makes sure that the casino has an advantage over the players in the long run. With that in mind, the next of our blackjack tips is one that can make a great difference for you.

• Set a budget and abide to that. Experienced blackjack players know that you need to set a losing and a winning limit before the first hand is dealt.

  Stop playing as soon as you hit one of them. If you lose and hit your lower limit, take a break and end your session.

  If you win and you hit your highest limit, collects your winnings and leave the table. You can continue to play if you want, but you'll need to set a new bankroll and establish new limits first.

• Don't miss the rules sheet. Trust me, there are more blackjack variations that you will ever play. All of them are based on the same core gameplay — but adopt rules that could break you if you aren't ready for them

  One of the best parts of playing online is that the rules are always available at the table. If you are not sure you know everything about the game you want to play, click on the 'info' icon and read the rules.

• Choose 3:2 over 6:5 payout. Different casinos have different payout policies - and not all of them are great for you.

  Make sure you read the payout table before you begin to play a hand of real money blackjack and always choose the games with the 'regular' 3:2 payout. These are a much smarter choice compared to the 6:5 payout blackjack games you find at some online casino sites.

Use Bonuses to Find Your Favourite Blackjack Strategy

Now that you know what are the best blackjack strategies, there is only one thing left: play a few games online to find out which one works best for you.

Here's what you should fo to get the best results:

1. Base all your decisions on the basic blackjack strategy
2. Pick the betting strategy that suits your bankroll
3. Follow all the blackjack strategy tips above to maximise your chances to win

To help you find the right games and the right betting strategy, we created a list of the best blackjack bonuses on the market.

All the sites you find in the list are licensed, pay winnings fast, and offer lots of blackjack games. Pick the ones with the best offers, register an account (it's always free), and sit at the blackjack table to start an epic battle with the dealer!