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The problem is that $N(x)/N$ is a probability density, not a probability. You can see that from the units: $N/\sigma$ has units "number per meter". You have to ask: How many balls will I find between $x_1$ and $x_2$? The expected number of balls is calculated from this density: $$ \bar{N} \approx N(x) \Delta x$$ if $\Delta x = x_2-x_1$ is small ...


In general if you have m out of n atoms on average of a certain composition, then if you take a sample of size N, you can approximate the distribution of atoms with a binomial distribution with mean $$mean = N \frac{m}{n}$$ and standard deviation $$std = \sqrt{N\left(\frac{m}{n}\right)\left(1-\frac{m}{n}\right)}$$ This is on the assumption that the atoms ...

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