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Answer by Eric for Quickly calculate the probability of 12 dice having sum less than 30

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Well, your approximation is correct though you can do things in a quicker way without requiring paper. One fact is that the variance of a uniform variable interval of length $L$ is $L^2/12$, so it’s a quick approximation to get that the variance of a die is around $6^2/12=3$. Combining this with a total mean of $12\cdot 7/2$ gives a threshold of $(42-30)/\sqrt{12*3}=2$ standard deviations so $(1-.95)/2=.025$. That is there’s approximately a 2.5% chance of the sum being less than 30.

If it hadn’t luckily been an integer number, you could still approximate the number with linear approximation between 68,95,99.7%, rounding upwards for concavity.


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