As part of my curriculum, we need to know how to approximate a binomial probability using the normal distribution but I can’t see why you would ever do this in real life. What scenarios would require this method?

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Normal distribution generally approximate a lot of things very well, and have nice mathematic property. The fact that normal distribution approximate a lot of things very well is justified by central limit theorem; and this theorem is applicable to binomial. When you have independent variables that are all normal, their vector is also normal, and all linear combinations of them are also normal; these are very nice properties that make normal distribution easier to calculate with.

With large values of n (the population), the binomial is very cumbersome to calculate.

Say you have 1000 throws of a dice and sum the face value of each throw. What is the likelihood of getting less than 2200 as the total?