I need help with a math equation to describe a scenario.

Here’s the scenario:

Picture a large piece of cheese. There are 100 mice released at the same time that compete for the cheese.

On average each mouse would get 1/100th of the cheese but if you looked at the distribution of mouse performance on a graph it would look like a bell curve with half the mice performing better than average and half performing worse.

If we run this test a hundred times the average mouse will still get 1/100th of the cheese and the distribution of performance will still be a bell curve but the shape of the bell curve will change. If we assume that the performance of each mouse is random during each run (no inherent mouse skill) the bell curve should become more “narrow” but the center of the curve will remain in the same location. Eventually, if we run this to infinity the bell curve should in essence become a vertical line on the graph at 0.01

Any ideas how to describe this in an equation?

To describe the scenario you've outlined mathematically, we can use concepts from statistics, particularly the normal distribution and the law of large numbers. Your scenario involves 100 mice competing for a piece of cheese, and you're interested in how their performance can be modeled and how it changes over repeated trials. Let's break this down step by step.

Understanding the Distribution of Performance

In your scenario, each mouse's performance can be thought of as a random variable. When you say that the performance follows a bell curve, you're referring to a normal distribution. In a normal distribution, most values cluster around the mean, with fewer values appearing as you move away from the mean in either direction.

Defining the Variables

Let's define some variables to help us formulate the equation:

• N: Total amount of cheese (let's say 1 unit for simplicity).
• M: Number of mice (100 in this case).
• X: Performance of each mouse, which we can assume is normally distributed.
• μ: Mean performance of the mice, which is 0.01 (1/100th of the cheese).
• σ: Standard deviation of the performance, which will change as we run the experiment multiple times.

Modeling the Performance Distribution

The performance of each mouse can be modeled as:

X ~ N(μ, σ²)

Where:

• μ = 0.01 (the average amount of cheese each mouse would get).
• σ is the standard deviation, which will decrease as you run more trials.

Behavior Over Multiple Trials

As you conduct this experiment repeatedly, the central limit theorem comes into play. This theorem states that as the number of trials increases, the distribution of the sample means will approach a normal distribution, regardless of the original distribution of the data. In your case, even if each mouse's performance is random, the average performance across many trials will still center around 0.01.

However, the standard deviation (σ) will decrease as you run more trials. This is because the variability in the average performance will reduce as you gather more data points. Mathematically, the standard deviation of the sample mean can be expressed as:

σ_X̄ = σ / √n

Where:

• σ_X̄ is the standard deviation of the sample mean.
• n is the number of trials (in this case, the number of mice).

Long-Term Behavior

As you run the experiment to infinity, the distribution of the average performance will become increasingly narrow, converging towards the mean of 0.01. In the limit, the distribution can be thought of as approaching a vertical line at 0.01 on the graph. This is a manifestation of the law of large numbers, which states that as the number of trials increases, the sample mean will converge to the expected value.

Final Equation Summary

In summary, the performance of the mice can be modeled using:

X ~ N(0.01, σ²)

And as you conduct more trials, the standard deviation of the average performance will decrease, leading to:

σ_X̄ = σ / √100

Ultimately, as the number of trials approaches infinity, the distribution of average performance will converge to a point at 0.01, represented as:

lim (n → ∞) P(X̄ = 0.01) = 1

This mathematical framework captures the essence of your scenario, illustrating how the performance of the mice converges over repeated trials. If you have any further questions or need clarification on any part of this explanation, feel free to ask!