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For 2 and 3 variables it's easy to "visualize" such problems using a two-dimensional and three-dimensional coordinate grid. My question is about algebraically solving those types of problems.

What's the probability that the sum of four uniformly picked values from the interval (0,1) is less than 3? (in other words, $w+x+y+z < 3$)

This WolframAlpha integral query gives the correct solution. I understand that it's equivalent if we replace $min(1, 3-w)$ and $min(1, 3-w-x)$ upper bounds with $1$ because $3-w$ and $3-w-x$ are always greater than 1:

enter image description here

However, what I don't understand is:

  • How to get to such integrals? If the question asks about a sum greater than something, by intuition the lower bounds would have to be changed somehow instead of the upper bounds. If there are multiple inequalities given, $min$ and $max$ functions would have to be used (inspiration taken from this answer)

I'd also appreciate if you can link any books/papers that approach similar problems in a not-so-advanced way

musava_ribica
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  • By analogy with $2$ and $3$ dimensions the $4-$volume the four numbers can occupy is a tessaract. The region where the sum is greater than $3$ is the equivalent of a pyramid cut off one corner with all sides $1$. In $2D$ the area cut off is $1/2!,$ in $3D$ the volume is $1/3!$ and in $4D$ it is $1/4!$ – Ross Millikan Oct 29 '23 at 20:08
  • How do you come to this integral? – user Oct 29 '23 at 20:08

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