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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.
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.
Let's define some variables to help us formulate the equation:
The performance of each mouse can be modeled as:
X ~ N(μ, σ²)
Where:
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:
σX̄ = σ / √n
Where:
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.
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:
σX̄ = 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!
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