# Why can we write any lottery as a convex combination of the degenerate lotteries?

I know that a degenerate lottery is a lottery that yields outcome $$n$$ with probability $$1$$ and I also know the definition of convex combination: given $$x_{1},x_{2}, \cdots ,x_{n} \in \mathbb{R}$$, a convex combination of these points is a point of the form $$\alpha_{1}x_{1}+...+\alpha_{n}x_{n}$$ such that $$\alpha_1+...+\alpha_n=1$$.

But I am quiet confused about why we can write any $$L=(p_1,...,p_n)$$ as a convex combination of the degenerate lotteries $$(L^1,...,L^n)$$. Can someone please explain this for me? Many thanks!

Pick any set of non-negative $$p_1,\dots,p_n$$ such that $$p_1+\cdots+p_n=1$$. The convex combination of degenerate lotteries $$L^1,\dots,L^n$$ with the $$p_i$$'s can be written as \begin{align} p_1L^1+\cdots+p_nL^n&=p_1(1,0,\dots,0)+\cdots+p_n(0,\dots,0,1)\\ &=\bigl(p_1(1)+p_2(0)+\cdots+p_n(0),\;\dots,\;p_1(0)+\cdots+p_{n-1}(0)+p_n(1)\bigr)\\ &=(p_1,\dots,p_n). \end{align} This is just a lottery $$L=(p_1,\dots,p_n)$$.