Probability of the sum of three random variables being less than 1 Given three Uniform random variables between 0 and 1, $(x,y,z)$, that are i.i.d., what is the probability $x+y+z < 1$?
 A: Your probaility is
$$\int_0^1\int_0^{1-x}\int_0^{1-x-y}1\,dzdydx=\frac{1}{6}$$
A: Does this help?
Histogram[Total@RandomReal[{0, 1}, 3] & /@ Range@1000000, 50]


With[{nn = 6*10^6}, Length@Select[Total@RandomReal[{0, 1}, 3] & /@ Range@nn, #  
< 1 &]/nn // N]

appears to tend towards $\dfrac{1}{6}.$
A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

A generalization for $\ds{k}$ variables is straightforward by using the Laplace Transform 'technique'. Hereafter, $\ds{\bracks{\cdots}}$ is an Iverson Bracket.

Namely, 
\begin{align}
&\bbox[10px,#ffe]{\ds{\int_{0}^{1}\cdots\int_{0}^{1}
\bracks{x_{1} + \cdots + x_{k} < 1}\dd x_{1}\ldots\dd x_{k}}}
\\[5mm] = &\
\int_{0}^{1}\cdots\int_{0}^{1}
\bracks{\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}
{\exp\pars{\bracks{1 - x_{1} - \cdots - x_{k}}s} \over s}\,
{\dd s \over 2\pi\ic}}
\dd x_{1}\ldots\dd x_{k}
\\[5mm] = &\
\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}{\expo{s} \over s}
\pars{\int_{0}^{1}\expo{-sx}\dd x}^{k}{\dd s \over 2\pi\ic} =
\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}{\expo{s} \over s}
\pars{1 - \expo{-s} \over s}^{k}{\dd s \over 2\pi\ic}
\\[5mm] = &
\sum_{n = 0}^{k}{k \choose n}\pars{-1}^{n}\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}{\expo{\pars{1 - n}s} \over s^{k + 1}}{\dd s \over 2\pi\ic} =
\sum_{n = 0}^{k}{k \choose n}\pars{-1}^{n}\bracks{n < 1}
\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}
{\expo{\pars{1 - n}s} \over s^{k + 1}}{\dd s \over 2\pi\ic}
\\[5mm] = &
\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}
{\expo{s} \over s^{k + 1}}{\dd s \over 2\pi\ic} = \bbx{\ds{1 \over k!}}
\end{align}
