Casinos Bonuses Theory

It's mathematically possible to have an edge over Casinos giving bonuses.

I won't explain how casino bonuses work, but it's a prerequisite to understand the Casinos Bonuses Theory.

Definitions

Let's start with the definitions:

D: Deposit.
b: Bonus percentage (most of the bonuses work with a percentage instead of a fixed amount, that's why I put it in the equation).
W: What you need to wager to clear the bonus.
RTP: Return To Player (%). Ex: RTP = 98%. Player plays 1000€. He can expect to get 980€ back.
B: Bonus amount.
n: Wagering requirement. How much times I need to wager the bonus amount to clear it.
P: Profit.
Volatility: There can be a high deviation of RTP for small size samples.
G: Gain constant, in percent.

Concept

We can make money with online casinos with bonuses. The only requirement is to have a positive gain constant.
Here is the gain constant formula:

G = 1 - n (1 - R T P)

The gain constant tells us how much we can expect to earn, as it's a percentage of the bonus. If $G = 0.2 (o r 20 %)$ , then I can expect to earn 20% of my bonus amount $B$ . If it's negative, I can expect to loose.

To put the odds on our side, we have to target games with the highest RTP possible and the lowest volatility to avoid randomness as much as possible.

Proof

\begin{array}{r} B = b * D \\ W = B * n \end{array}

In order to make a profit, we should in the end win more than the bonus amount. What we have to wager is W, what we will get back is $W * R T P$ (which is a percentage). What we keep is what we get with the RTP, minus what we wager, plus the bonus amount. And it should be positive if we want a profit, else it's a loss.

\begin{array}{r} W * R T P - W + B > 0 \\ W - R T P * W < B \\ 1 - B / W < R T P \\ R T P > 1 - \frac{1}{n} \end{array}

We can now introduce a "gain constant" $G$ . It represents the percentage of the bonus I'm winning (or losing). If the constant is positive, I'm winning money.

G = 1 - n (1 - R T P)

We can even calculate the profit:

P = B * G

Example

I deposit 100$ and have a 100% bonus. The wager requirement is 40.

\begin{array}{r} R T P > 1 - \frac{1}{40} \\ R T P > 97.5 \end{array}

I find a game with a 99% RTP.

\begin{array}{r} G = 1 - 40 (1 - 0.99) \\ G = 0.6 \\ G > 0 \end{array}

I can calculate my expected profit.

\begin{array}{r} P = 100 * 1 * 0.6 \\ P = 60 $ \end{array}

Now, let's say I play instead to a game with a 95% RTP.

\begin{array}{r} G = 1 - 40 (1 - 0.95) \\ G = - 1 \\ G < 0 \end{array}

I won't have any profit since G < 0. I can calculate my loss.

\begin{array}{r} P = 100 * 1 * - 1 \\ P = - 100 $ \end{array}

I have a profit of -100$ (a loss).

All this is theoretical. Depending on your luck, it may be worse or better. That's why in my opinion it's better to target games with a low volatility. This way, you reduce the impact of randomness.

Graphs

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