Let $a,b,c,d,e$ be drawn independently and uniformly at random from $[0,1]$. What is the probability that $a+b+c>d+e$?

Considering a unit hypercube, we want the volume "below" the plane $a+b+c-d-e=0$. The corners that are "above" the plane are $$(0,0,0,1,0),(0,0,0,0,1),(0,0,0,1,1),(1,0,0,1,1),(0,1,0,1,1),(0,0,1,1,1).$$ How can we compute the volume of this part of the hypercube?

  • $\begingroup$ The $4-\operatorname{volume}$? $\endgroup$ – user142198 Feb 27 '15 at 3:00

Notice if $d, e$ are random variables uniform over $[0,1]$, so do $\tilde{d} \stackrel{def}{=} 1 - d\;$ and $\;\tilde{e}\stackrel{def}{=} 1 - e$.
We have $$a + b + c > d + e \quad\iff\quad a + b + c + \tilde{d} + \tilde{e} > 2$$ For any $p = (x_1,x_2,x_3,x_4,x_5) \in \mathbb{R}^5$, let $\;\ell(p) = x_1 + x_2 + x_3 + x_4 + x_5$.
The probability $\mathcal{P}$ we seek is equal to:

$$\mathcal{P} = \verb/Volume/\big\{\; p \in [0,1]^5 : \ell(p) > 2 \;\big\} = 1 - \verb/Volume/\big\{\; p \in [0,1]^5 : \ell(p) \le 2 \;\big\}\tag{*1}$$

Let $e_1, e_2, \ldots, e_5$ be the standard bases of $\mathbb{R}^5$ and $\Delta_p(\lambda)$ be the simplex $$\big\{\; q \in \mathbb{R}^5 : q_i \ge p_i, i = 1\ldots5\;\text{ and }\; \ell(q-p) \le \lambda \;\big\}$$

The polytope on the right of $(*1)$ can be constructed by cutting 5 small simplices based at $e_i$ from the simplex $\Delta_0(2)$. More precisely, up to a set of measure zero, we have: $$\big\{\; p \in [0,1]^5 : \ell(p) \le 2 \;\big\} = [0,1]^5 \cap \Delta_0(2) \approx \Delta_0(2) \setminus \left( \bigcup_{i=1}^5 \Delta_{e_i}(1) \right)$$ This leads to$\color{blue}{^{[1]}}$

$$\mathcal{P} = 1 - \frac{2^5}{5!} + 5 \left(\frac{1^5}{5!}\right) = 1 - \frac{32 - 5}{120} = \frac{31}{40}$$


  • $\color{blue}{[1]}$ In general, in any dimension, the volume of the simplex $\Delta_0(\lambda)$ inside the unit hyper-cube is given by following formula: $$\verb/Volume/\big( [0,1]^n \cap \Delta_0(\lambda) \big) = \sum_{k=0}^{\lfloor\lambda\rfloor} (-1)^k \binom{n}{k} \frac{(\lambda-k)^n}{n!}$$ One can prove this formula using inclusion exclusion principle. Look at my answer for a similar question in $\mathbb{R}^3$, it has more explanation of the underlying ideas.

$$\int_0^1 da \int_0^1 db \int_0^1 dc \int_0^{{\rm min}(1,a+b+c)} dd \int_0^{{\rm min}(1,a+b+c-d)} de = \frac{31}{40}=0.775$$

  • $\begingroup$ Could you please show how to do this integral? The $\min$ part makes it complicated. $\endgroup$ – python55 Feb 27 '15 at 3:27
  • $\begingroup$ As a warmup, consider the simpler problem $a+b>c$. The integral is $\int_0^1 da \int_0^1 db \int_0^{{\rm min}(1,a+b)} dc$. For the last integral you consider two cases: $a+b<1$ and $a+b\ge1$, each of which affects the limits for the $db$ integral (in the first case, $b$ runs from $0$ to $1-a$; in the second case, $b$ runs from $1-a$ to $1$.) You end up with two triple integrals and no "min"s in sight. (The answer here is 5/6, as a check.) $\endgroup$ – Tad Feb 27 '15 at 3:50

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