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An Inside Look at the Mechanics of Betting Progressions

29 August 2000

The average casino gambler seems to be permanently bonded to the notion that by sizing his next bet according to the last outcome, he'll win more money than he'll lose, even though he'll lose more bets than he'll win. What a neat trick. Can it work? Well, if you're analyzing it from a purely mathematical standpoint, it cannot.

To illustrate, let's take a look at a popular progressive betting system known as the "1-2-3-5 step-up" progression. Here's how it works:

You start out betting 1 unit and stick to a 1-unit bet if you lose. If you win, you progress to a 2-unit wager and if you win again, progress to 3-unit wager. If you win your third consecutive bet, you progress to a 5-unit wager and stay there as long as you keep winning. You always revert to a 1-unit wager immediately following any loss on any step of the progression.

Suppose you were gambling with your buddy on a 50-50 proposition, such as a coin flip. If you were flat betting (same size bet on every flip), you'd stand to break even over the long haul since you should win as many flips as you lose. But how would you make out if you used the 1-2-3-5 progression?

On the surface, the system looks like a winner because when you win four bets in a row you gain 11 units (1+2+3+5) - but when you lose four straight it costs you only four units (1+1+1+1). So far, so good. What happens when you win three and lose one, or win one and lose three? How about when you go two and two? To answer these questions, take a look at the chart below.

WWWW +11   WWWL +1   WWLL -1   WLLL -3   LLLL -4
      WWLW +1   WLWL -2   LWLL -3      
      WLWW +2   WLLW -1   LLWL -3      
      LWWW +5   LWWL -1   LLLW -2      
            LWLW -1            
            LLWW +1            

This chart shows every possible outcome for four bets. As you can see, your results for a particular sequence of outcomes depend not only upon how many wins and losses you have, but also upon the order they come in. Notice too that the chart contains as many total wins as losses (32 of each) which it should, since a coin flip is a 50-50 proposition. But now I have to ask you a very important question:


The answer is, they are all equally likely. "WWWW" is just as likely as "LLLL" or even "WLWL" for that matter. It's true that it's much easier to win two and lose two than it is to win all four or lose all four. But there's only one way to win them all or lose them all, and there are six ways to go two and two (The entire center column!). Any specific sequence of outcomes is just as likely as any other.

The result each sequence would produce is shown to its right.  Add up all the sequences that produce a gain and you get 21 units won. Now add up all the sequences that produce a loss and what do you get? You get 21 units lost.

Let's stop right here and think about this. We know that all 16 sequences are equally likely to occur. Some of them produce a loss while others yield a gain. But since they're all equally likely, where would we be if we bet on four flips of a coin 16 different times and got every different sequence once each? We'd be dead even, wouldn't we? Well, that's not an edge!

Can you see that when you bet on four flips of a coin using this betting progression, it's like picking a piece of paper out of a hat that contains 16 slips of paper in all, each one with a different four-flip sequence written on it? Since all the pluses add up to the same number as all the minuses, you're still an even money shot (in a 50-50 game). The only unbalanced part is that you have a chance to score a bigger winner than your maximum loser -- but there are more losing slips of paper in the hat.

That's just what positive betting progressions do. They gather the bulk of your wins into a condensed area -- your streaks. And they spread out your smaller losses across a broader range (notice that going 2 and 2 usually costs you money). But your overall edge (or disadvantage) remains unchanged.

I need to emphasize here that this is not just some phenomenon that occurs with the 1-2-3-5 progression alone. On the contrary, the exact same thing is true of any progression -- positive, negative or regressive. Go ahead, try out your own home-spun concoction on the win/lose chart right now. You'll always find that the total units won and total units lost add up to the same number.

It also doesn't matter whether you play your progression for 4 bets, 40 bets or 4000 bets. That's because for any number of bets there are just exactly so many possible win/lose sequences that they can come in. With 4 bets, there are just 16 ways things can come out. With 10 bets, there are 1024. And with 100 bets, the answer is a big honkin' 31-digit number. But it doesn't matter. If you laid all the sequences out on paper and added together the gains from all the winning sequences, they will always equal the total losses sustained by all the losing sequences. And when you gamble, you'll just be drawing one of those sequences out of that big hat whose pieces of paper all add up to zero.

Drawing a sequence that contains a winning streak scores a memorable win. If your sequence contains a losing streak, you lose a lesser amount. But here's the part nobody seems to realize -- if your sequence contains exactly as many wins as losses, you'll most likely lose a little bit. That's because going WLWLWL, etc, costs you money with a positive betting progression. After you add your moderate losses from your losing streaks together with your minimal losses accumulated through the prolonged chops, the number will equal the larger amount won during your winning streaks once all phases have occurred. Your progression has merely redistributed the gains and losses without changing the bottom line. So, what's it all mean? It means that


So far, we can see that, advantage-wise, nothing changes when you apply your favorite betting progression to the flip of a coin, the roll of the dice or the spin of a wheel. But as of late, the furor has re-ignited over whether progressive betting has merit in the game of blackjack.

First, understand that betting progressions can't work in an unbiased process because once all sequences occur as often as they're supposed to. But what if there was something preventing all sequences from occurring as often as they're supposed to? Just what if the combinations of two wins and two losses from the center column in the chart somehow occurred less frequently than they were supposed to, and the streaks occurred more? If that were true, then there would be more + 11's and -4's, and betting progressions would, in fact, produce a net gain!

What could ever skew the natural distribution of equally likely sequences? Some people believe card clumping causes just that very thing to occur in blackjack. Gamblers have been saying for twenty years that shoe games seem to run more streaky than handheld games. For that reason, some blackjack players insist their own personal betting progression is a winning formula.

To determine whether streaks actually do occur more often than predicted by their mathematical probability would require a mammoth hand-dealt experiment. A computer would only force the cards into random order during the shuffle. Every next card would then come randomly, and the frequency of streaks would conform to the math. If shoe-dealt games do run more streaky, there's something at work in the material game forcing wins to follow wins and losses to follow losses at a more consistent rate than random. I see no way to prove this other than by amassing a huge manually dealt sample.

Since I often practice blackjack at home, I decided to start tracking my own streaks. Dealing from a six-deck shoe, I recorded the number of times I won or lost three or more consecutive hands over the course of 2000 hands played. The straight mathematical probabilities for winning, losing and pushing a hand are 43%, 48% and 9%, respectively. Based upon those percentages, there should have been 92 occasions on which I won at least three in a row, and 115 occurrences of three or more straight losses. The actual tallies were 88 such winning streaks (rather than 92) and 116 losing streaks (rather than 116). These results fall awfully close to the mathematically predicted outcomes.

Admittedly, there are two things wrong with my experiment. First, 2000 hands is by no means an eternity. Still, both kinds of streaks occurred an appropriate number of times. That leaves only two conceivable conclusions, Either streaks really don't occur any more often than they're supposed to, or this was an uncommonly choppy 2000 hands.

The second problem is that I played these hands "head up" against the dealer. Proponents of progressive betting often cite crowded tables as the cause of streakiness, espousing that the somewhat consistent drawing/standing strategy used by the masses tends to cause the cards to clump into strings of consecutive highs and consecutive lows. This, they say, is not totally obliterated by a manual shuffle and causes the players to win more consecutive hands through high card clumps, then lose more successive hands through low card clumps. Personally, I don't understand why this should be so since, if you think about it, the cards will come out in a different order on the next shoe even if they weren't shuffled at all.

So, what's the bottom line on progressive betting systems? This much we know for sure:

  • If streaks occur merely as often as they're supposed to in multi-deck play, there's just no way for betting progressions to alter the player's expectation. Blackjack would be locked into the same mathematical constraints with craps, roulette and a coin flip.
  • So many typical players instinctively use some sort of progressive approach to their betting that there would be lots of long-term winners at blackjack, and there are not.

In the end it all comes down to the randomness question. Until it's determined positively whether shoe-dealt blackjack is more streaky than its theoretical probability dictates, the contention that progressive blackjack wins will remain a hotly debated controversy. Me? I seriously doubt it works.

Fred Renzey
Fred Renzey is a high-stakes, expert poker player. On a daily basis he faces--and beats--some of the best players in the country in fierce poker room competition. Now for the first time, Renzey offers his perceptive insights on how to play winning poker. For Fred's 13-page blackjack booklet "Ace/10 Front Count", send $9 to Fred Renzey, P.O. Box 598, Elk Grove Village, IL, 60009

Books by Fred Renzey:

Blackjack Bluebook II

> More Books By Fred Renzey

Fred Renzey
Fred Renzey is a high-stakes, expert poker player. On a daily basis he faces--and beats--some of the best players in the country in fierce poker room competition. Now for the first time, Renzey offers his perceptive insights on how to play winning poker. For Fred's 13-page blackjack booklet "Ace/10 Front Count", send $9 to Fred Renzey, P.O. Box 598, Elk Grove Village, IL, 60009

Books by Fred Renzey:

> More Books By Fred Renzey