In movies, we often see a gambling method in casinos where players bet on whether the number will be odd or even. If they lose, they lose the amount they bet; if they win, they gain an amount equal to the amount they bet. It is said that there is a "guaranteed win, no loss" strategy for this betting method, known as the "double bet" method.
The specific strategy of the double bet method is: the first time, bet a minimum amount of a yuan. If you lose, double your bet to 2a yuan the next time. If you still lose, double your bet to 4a yuan. This continues, doubling your bet each time you lose until you win. Then, start over with the minimum bet and repeat the process. Assuming you lose n consecutive bets before winning, you will lose a total of a + 2a + ... + 2^(n-1)a = (2^n - 1)a yuan, and the next bet will win you back 2^n a yuan. Therefore, you will always win a total of a yuan in one round! It seems that as long as you keep gambling, you can guarantee a win without losing. This seems like a perfect foolproof winning strategy.
But is that really the case? In fact, many people who know about this "double bet" method have gone to the casino to try their luck. However, the result is that they usually end up losing everything they have. In other words, this "guaranteed win" method doesn't work. Why is that? The reason is that even if the gambler can win every time, they can only win the minimum amount of a yuan. Even if they win n times in a row, they will only win na yuan, which is an arithmetic progression with a slow growth rate. However, once they start losing, the amount they lose doubles at an incredibly fast pace. As long as the gambler's capital is limited, they may lose all their capital before they have a chance to win. For example, if a gambler brings 10,000 yuan in capital and bets with a minimum of 10 yuan, if they lose 10 consecutive times, they will lose 10 × (2^10 -1) = 10,230 yuan. Therefore, even if they win a little money at the beginning (which won't be much because the winning amount only grows in an arithmetic progression), they will basically lose it all.
Some readers may think that although there is a possibility of consecutive losses, the chance of losing 10 times in a row is very small. The probability of losing once is 1/2, the probability of losing twice is (1/2)^2 = 0.25, and the probability of losing 10 times in a row is only (1/2)^10 ≈ 0.001. Such a small probability is unlikely to occur. However, if the gambler believes that the "double bet" method guarantees a win, they will keep gambling, and the chance of losing 10 times in a row will be much higher than imagined. Calculations show that if they bet about 1,500 times, the probability of losing 10 times in a row will exceed 1/2, which means they will lose all their 10,000 yuan in capital with a probability greater than 1/2. Moreover, the more times they bet, the higher the chance of consecutive losses will be. If they bet 5,000 times, the probability of losing 10 times in a row will exceed 90%.
In general, as long as the gambler keeps gambling, the probability of losing 10, 15, or 20 times in a row will increase as the number of bets increases. Therefore, no matter how much capital a gambler has, as long as it is limited, they will eventually lose it all.