The Maximum Boldness Theorem in gambling

The maximum boldness theorem is a basic law of gambling that comes into play when you're up against an edge. It is a complex mathematical theory which, simply put maintains that when starting with a given stake in a casino game, you'll have the best odds of reaching a specific level of returns by placing the maximum possible bet consistent with achieving your objective in the fewest possible wagers.

So, this theorem proposes that the more money you wager during a short gambling session, whether you reinvest your immediate wins or compensate for your losses by using additional cash from your bankroll, the more effect you'll have on the house edge in various casino games.

Let's examine this theory by way of an example, on an even money bet with a 49% chance of winning and 51% chance of losing, which gives the house a 2% advantage in the game.

For the sake of example, let's say you have a €1000 stake and a €1000 earnings goal, if you wager the full €1000, your odds of winning are 49%. If instead you decide to wager €500 and you win, and wager another €500 and win again, you'll have earned €1000 and reached your goal, but you will have to win twice in a row to achieve that, and the odds of that will be a mere 24% (49% multiplied by 49%). In staying with €500 bets, you could wager €500, and lose the first bet and still be able to meet your earnings goal if you win the next three bets, but the chances of that happening are 6%. Or you may need to place four bets to reach your goal, if you win your first bet, lose the next and then win the two wagers after that - but again only with a 6% chance of this happening, which in keeping the prior two scenarios in mind will give you a 36% chance of reaching your goal (24 + 6 + 6).

If you played longer combinations in €500 increments, the best you could hope to achieve is a 48% chance of winning, which is not as good as the odds you'll achieve when applying the maximum boldness strategy and wagering your full €1000 on a single bet. This is because as your wagers decrease in value, and increase in number, your chances fall. Let's take a look at this in table format for casino games with a 2% and 5% house edge.

The odds of double a €1000 bankroll in an even money game with 2 and 5% respectively when placing wagers of varying sizes:

Wager Size Odds of Success
(2% house edge)
Odds of Success
(5% house edge)
€1,000 49% 48%
€500 48% 45%
€250 46% 40%
€100 46% 27%
€50 40% 12%
€25 17% 2%
€10 2% Less than 1%

Of course, you may be wondering if the maximum boldness theorem applies to bets paying more than even money. Let's examine this query by way of example.

If a proposition bet pays 10:1 and the casino has a 2% advantage, the probability of winning would be 8.9%. The odds of doubling a €1,000 stake rather than busting out completely when betting €100 per round is just under 49%. If you make wagers of lower value, let's say €25, your odds drop to 46% and on €10 wagers your chances decrease to 40%. As you can see, the maximum boldness theorem applies regardless of the payout ratio offered on a specific bet.

That said though, it also tells us that the odds of reaching our earnings goal before expending our casino bankroll drops more slowly when decreasing the value or our bets in the face of bets paying out above even money.

There are of course exceptions to this theorem, for example if you decide to wager your entire stake to double your money in craps, the maximum boldness theorem will suggest that you go for broke on a single roll of the Dice and wager on Don't Pass. In doing so you would have a 49.3% chance of winning. If table limits prevent you from doing this, there may be times when your odds would be better by placing less than the maximum possible bet and keeping some of your bankroll in reserve to lay Odds if the shooter in the game establishes a point. This is due the fact that Odds reduce the edge over the course of the game. This is similar to a game of Blackjack where you don't bet the maximum and instead reserve funds to split or double and improve your chances of winning.

Of course, very few casino players ever gamble with strict loss limits and earning goals in place to the extent that the maximum boldness theorem would apply to their game play. This is largely due to the fact that most players want to play for fun and profit, and placing multiple bets is always more fun than placing a single bet that could go either way and then end the fun there.

Still if you find yourself looking for a good strategy and have the discipline to play accordingly, it could work for you.