The maximum boldness and minimum boldness strategies are two different ways to double your money in a casino, depending on whether you’re playing a negative expectation game or a positive expectation one.
Maximum boldness is used in negative expectation games – the ones where the casino has an advantage. You should try to double the money in a single bet. For example, in roulette it would mean betting everything on an even-money (1:1) wager. The chances of winning, which equals doubling the money, are about 48%. If you place a lot of small bets instead, your chances decrease. Let’s say you play a game in which you have 49% chances of winning and 51% of losing, you have $1000 and want to win another $1000. After the first even-money bet, your chances are 49%. But if you want to place two $500 bets instead, the chances of hitting $1000 drop to 24%.
Minimum boldness is used in positive expectation games, where the players has advantage over the casino. For example in blackjack, with perfect strategy and card-counting, the house edge can be -0.5%. In this case the best way to double your money is to place the smallest bets possible as many times as possible. This way you’re milking the house edge, just like the casino would be doing in a negative expectation game if you placed many small bets. If your chances of winning were 51% and your chances of losing were 49%, betting all the money on the first bet is actually very risky compared to taking it slow and having much bigger chances of doubling it.