A game has multiple winnigs over the expected limit. The maximum of them is 10000. How can i as a game designer calculate and make sure with 95 % confidence that the game will not go over the new calculated limit.

To ensure that a game does not exceed a certain threshold for winnings with 95% confidence, you can use statistical techniques, particularly focusing on confidence intervals and probability distributions. Here's a step-by-step approach:

### 1. **Define the Parameters of Your Game’s Winnings:**
   - **Maximum Winnings:** The maximum observed winning is \$10,000.
   - **Mean (Expected Value) Winnings (µ):** This represents the average winnings, which you may have from past data.
   - **Standard Deviation (σ):** This measures the spread of the winnings around the mean, which you can compute from your data.

### 2. **Choose a Distribution:**
   Depending on the nature of your game, the winnings might follow a certain distribution. Common choices are:
   - **Normal Distribution:** Often used if the winnings are continuous and symmetrically distributed.
   - **Log-Normal Distribution:** If the data is skewed (for example, if there are many small winnings and a few large ones).
   
   For simplicity, I'll assume a normal distribution for now.

### 3. **Calculate the 95% Confidence Interval:**
   The 95% confidence interval for your winnings can be calculated using the formula for a normal distribution:
   
   \[
   \text{Confidence Interval} = \mu \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
   \]
   
   Where:
   - $ Z_{\alpha/2} $ is the z-score for a 95% confidence level, which is approximately 1.96.
   - $ \mu $ is the mean winnings.
   - $ \sigma $ is the standard deviation of the winnings.
   - $ n $ is the number of winnings data points.

### 4. **Set the New Winning Limit:**
   - Once you have the confidence interval, set the upper bound (the mean + 1.96 standard deviations) as your new limit. This ensures that 95% of the winnings will stay below this limit.
   - Alternatively, if you don’t want to use the confidence interval approach, you can set the winnings threshold such that the probability of exceeding it is less than 5%. In a normal distribution, this can be done using the inverse of the cumulative distribution function (CDF), also based on the mean and standard deviation.

### 5. **Simulate the Outcomes:**
   - If you have access to your game's data, you can simulate winnings and see how often the maximum winnings exceed the calculated 95% confidence limit. This will help verify that your theoretical calculations match the practical outcomes.

### Final Approach:
   - Calculate the mean ($\mu$) and standard deviation ($\sigma$) of the winnings.
   - Use the z-score (1.96 for 95%) to calculate the upper limit of the winnings based on your confidence level.
   - Adjust the game’s payout mechanics to ensure the payouts stay within this limit.

Would you like me to go through a specific calculation or a different approach for a custom distribution?

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Here are 5 questions for further exploration:
1. What is the average or expected value of winnings for this game?
2. What distribution do the game’s winnings follow (normal, log-normal, etc.)?
3. How much historical data do you have to calculate standard deviation?
4. Would you like to explore a simulation to validate the calculations?
5. How frequently are winnings near the current maximum of \$10,000?

**Tip:** Simulating your game’s payouts using tools like Monte Carlo methods can help ensure that large winnings remain within safe limits.

A game has multiple winnings over the expected limit. The maximum of them is 10000. How can I as a game designer calculate and make sure with 95% confidence that the game will not go over the new calculated limit?

Learn how to calculate a 95% confidence interval to ensure that a game's winnings do not exceed a set limit, using statistical methods. This solution covers key concepts like confidence intervals, standard deviation, and normal distribution, and provides a step-by-step guide for game designers.

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