#4: Double the bet?

I guess most of the people, if not all, know a method of gambling that GUARANTEED a positive return, that is, you would just need to double your bet after losing. Try to think about it, when you bet a dollar, and if you lose, you bet two dollar, and if you lose, you make it four, eventually you will win for a round and you get back all the money that you loses, plus an extra dollar to your boost! Sounds good?

I am here to analyse the situation for you, let's make a few assumption to make our life simpler:

1. Initial bet is $1. (The amount doesn't matter anyway since we are only doubling after we lose)

2. Probability of winning is 50%. (This is just implying that the game we play is a zero-sum game, usually in casino there will be house advantage, which means a disadvantage for the players. So here let's assume we are playing the game of guessing big-small)


Now let's make a scenario for the method, it is theoretical but is definitely useful. Try with the easiest case, say, I have only 3 dollars and I want to try using the method as described above:

So I start by placing 1 dollar bet on the game, so if I win, I gain 1 dollar, and if I lose, I would place a two dollar bet now and hope that I win, if I win, I gain 1 dollar, and if I lose, there's the end for me. It is easy to illustrate it through a tree diagam, but I would use a table in my case.

Number of Bet | Probability of winning | Gain
#1 | 0.5 | $1
#2 | 0.25 | $1

In this case, the probability of losing is 0.25 which correspond to losing two round consecutively and the amount lost is $3.

So in this case, there is chance that I lose all my money!!

Let's calculate the expected value and variance of this case:
Expected value = 0.5 + 0.25 + 0.25(-3) = 0
Variance = 0.5 + 0.25 + 0.25(9) = 3

So basically I am just playing a zero-sum game, which is not different whether or not I am using the bet-doubling method. The variance here is used to measure the amount of risk (deviation of mean) of the game using this method.

For the following part, it will be a little more mathematic based and simple calculations are skipped:

So let's generalised the analysis, assume that my initial total money is $m dollar, and I still start the bet with $1, the table of analysis will now be:

Number of Bet | Probability of winning | Gain
#1 | 0.5 | $1
#2 | 0.25 | $1
#3 | 0.125 | $1
.
.
#n | (0.5)^n | $1

Probability of losing will now be (0.5)^n, and the amount that I might lose will be m = (2^n - 1). These amount can be calculated using the sum of geometry progressive, it is simple and straightforward so it won't be shown here.

Expected value = 0.5 + 0.25 + .... + (0.5)^n - (2^n - 1)(0.5)^n = 0
Variance = 0.5 + 0.25 + ... + (0.5)^n + (2^n - 1)^2 * (0.5)^n = (2^n - 1) = m

So we are still playing a zero sum game no matter what the n is (n here can be interpreted as the total number of bet you can even if you lose all the rounds). The variance is high here (equal to the money that I hold), this implies that the method is more risky than we just play the game once with a dollar.

So what is the conclusion drawn from the statistic analysis? It suggests that for a normal person, the method shouldn't be used as that would only increase the risk of losing more money. However, for a person who LOVES risk, he or she can go ahead.

Usually, the people who advocates this method would defend that doubling the bet method should only be used when a person has almost infinite money, well, if you carry on the similar analysis with n = infinity, you would find out that the expected value is 1 and the variance is zero, so it is CERTAIN that we can get a positive return of one dollar!! BUT, that is only when you have inifinite amount of money, no matter how rich you are, the money you have is definitely finite, and my analysis has shown that it is a zero-sum game with high risk (that you can finished your whole property). Besides, if a person is rich enough, then there is little point in winning more money as that only changes the figure of the wealth the person has.

So my personal advice is that, do not try playing in casino using this method as it is risky, it is even better if you stop gambling! The truth is, casino usually has advantage which suggests that probability of winning is lower than 50%.