Data Analytics Math

The Roulette Strategy

Why it will make you go bankrupt, mathematically

Wouldn’t it be nice if there was a strategy to guarantee a profit in gambling? Probably not, but I am not here to philosophize about it.

So what I am here for?
I am here to discover a gambling strategy, and show why it is simply not working mathematically — so you will be able to remove it from your “gambling strategies” folder, I hear everyone have those.

Before we go over the strategy, let’s get aligned with what are the features of a roulette (at least for the sake of this article).

  • Roulette has 37 slots ranging from numbers 0 to 36
  • 18 slots are red
  • 18 slots are black
  • The 0 slot isn’t colored

The Strategy

The strategy is fairly simple and easy to follow

  1. Place 1$ on red
  2. If you won, repeat step 1
  3. If you lost, bet again with a doubled amount
  4. If you win continue to step 1; else repeat step 3

The intuition behind this strategy is that after every losing streak if you win the next bet you cover your losses.
For example, if you just lost 3 times in a row, meaning that you lost 1$+2$+4$ = 7$, your next bet would be 8$ and if you win this bet you’ll have a profit of 1$.

Sounds like a winning strategy? think twice.

This Strategy Sucks

Before we dive into the maths, I want to show you graphically why this strategy won’t work in the long run.
I generated a few histograms from a martingale simulation that I came across while reading a blog post from everlastingbits — which revealed this topic for me.

Using his simulation, I generated two histograms.
I ran 5000 simulations for 5 years (simulated time) with a varying amount of initial amount of cash.

Running 5000 simulations for 5 years with 10k$ initial investment resulted in

To understand the graph we need to define what is a ‘profit ratio’.
I defined it as the total money you have at the end of the simulation divided by the amount of money you started with — for example if at the end of the simulation you doubled your money the profit ratio would be 2.

Analyzing the above histogram, we can see that if you come to the casino to bet every single day for 5 years with an initial 10k$, you are most likely to go back home with ~9.95k$ — not the results you were hoping to achieve.

I don’t know about you but 10k$ is a bit too much for me to bet on, so let’s check how good this strategy is when I will casually bet with 100$

Running 5000 simulations for 5 years with 100$ initial investment resulted in

So as you can see, by the end of these five years of diligently applying this strategy, on average you will lose around 13$.

To conclude this section, here’s the histogram of profit ratios when starting with 100$, 1000$, 5000$, and 10,000$

Why This Strategy Sucks?

Short answer: you don’t have an infinite amount of money to bet on.

Let’s try to understand this sentence by taking an example.
I am going into the casino with 50$, and this is my unluckiest day by far.
I am betting 1$, losing, doubling the bet, losing, and so on.
I can only allow myself 5 losses in a row, why?
1$+2$+4$+8$+16$ = 31$
By the time that I lost 5 rounds in a row, I am left with 19$, which is not enough in order to double my last losing bet. In this scenario, I just lose 31$ and go back home.

In general, in step of a betting round, my bet would be 2ᵏ⁻¹, and the total loss accumulated in these k steps is

With this in mind, let’s notate the amount of our initial cash as N, whenever we have to bet more than we have left, meaning when 2ᵏ > N we are done for.
And that will happen, as soon as step k = log₂N(*).

But how likely is it for me to lose so much in a row?
Since we have 18 red slots out of 37, our chances of losing each time are 18/37, and since the betting rounds aren’t dependant, we can state that the probability of losing k rounds in a row is (18/37)ᵏ.

We can do some hand waving persuading ourselves that (18/37)ᵏ = (1/2)ᵏ when k approaches infinity, and come to the conclusion that the probability of losing k times in a row is actually

Using (*)

This result simply tells us that our chances of losing decreasing as we start with a bigger amount of cash.
When you have an infinite amount of cash to bet on, you will know for sure that you can make a profit out of this strategy, but if you do, what’s the point?

As a final note, I want to give credit one more time to everlastingbits for his useful simulation. if you wish to check out the code for the simulation it is available in his blog post.

Shameless Self-Promotion

My girlfriend and I recently opened a greeting card shop together.
In case a loved one has a birthday soon or you need some Christmas cards in advance, we got you covered.

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