Here's how an AI works

 What an AI says worth as much as the verification you can make. So, it's exactly the same as talking to Donald trump.

Example:

1. Crash Distribution: The game is programmed to "crash" at a certain point based on a probability distribution. This distribution can be carefully tuned to achieve a specific RTP.

   • Expected Value: If the game crashes at a random multiplier, $X$, the expected value of $X$ is a key factor in determining the RTP. If $f(x)$ is the probability density function for the crash multiplier, then the expected value $E[X]$ is calculated as: $$E[X] = \int_1^\infty x \cdot f(x) \, dx$$

2. Player's Expected Cash Out: If a player cashes out at a multiplier $M$ with probability $P(M)$ before the crash, the expected payout for the player is:

   $$E[\text{Payout}] = \int_1^\infty M \cdot P(M) \, dM$$

   The RTP would then be the ratio of the player's expected payout to their bet:

   $$\text{RTP} = \frac{E[\text{Payout}]}{\text{Bet}}$$

3. House Edge and RTP: The game can be designed such that the probability distribution $f(x)$ (which influences where the game crashes) ensures that, on average, the game returns a certain percentage (the RTP) of all bets to players. The house edge is simply $1 - \text{RTP}$.

Example: Exponential Distribution in Crash Games

Suppose the crash multiplier follows an exponential distribution, which is commonly used because it's memoryless and fits the "random stop" nature of crash games.

If the crash occurs according to an exponential distribution with parameter $\lambda$, the probability density function is:

$$f(x) = \lambda e^{-\lambda x}$$

The expected value $E[X]$ for an exponential distribution is $1/\lambda$.

To achieve a specific RTP, say 95%, the game designers would tune $\lambda$ such that the expected payout corresponds to this RTP. If the average multiplier at which players cash out is, say $M$, then the RTP can be calculated as:

$$\text{RTP} = M \times \lambda \times e^{-\lambda M}$$

The value of $\lambda$ is selected to ensure the desired RTP.

Conclusion

The RTP in crash games, like in any other casino game, is a measure of the average return to the player over time. The crash point is governed by a probability distribution, and the game designers carefully tune this distribution to achieve the desired RTP. The mathematics involves understanding expected values, probability distributions, and their interplay with the payout mechanics.

This approach ensures that while the game may seem random and unpredictable, over the long run, the casino maintains a consistent edge, just as with slots or other gambling games.

You said:

"a multiplier 𝑀 M with probability 𝑃 ( 𝑀 ) P(M)" that makes no sense to me. The M is what I choose.

ChatGPT said:

You're correct to point out that the way I described the multiplier $M$ and the probability $P(M)$ was unclear. Let me clarify that.

Clarification of Multiplier and Probability

In a crash game, as a player, you decide when to cash out at a multiplier $M$. This decision is under your control, so there's no probability $P(M)$ associated with your choice of $M$.