P(k=n)=\int_0^1dx_1\int_0^{1-x_1}dx_2\cdots
\int_0^{1-(x_1+\cdots+x_{n-2})}dx_{n-1}\int_{1-(x_1+\cdots+x_{n-1})}^1dx_n=
\int_0^1dx_1\int_0^{1-x_1}dx_2\cdots
\int_0^{1-(x_1+\cdots+x_{n-2})}(x_1+\cdots+x_{n-1})dx_{n-1}.