<p>Consider an equilateral triangle of side length <span class="math-container">$\frac{2}{\sqrt{3}}$</span>. If we pick any point inside the triangle and drop perpendiculars to the three sides of the triangle, then the length of these perpendiculars can represent the probabilities associated with three outcomes in a given lottery. See the graph below and verify that <span class="math-container">$p_1+p_2+p_3=1$</span></p>
<p><a href="https://i.sstatic.net/rBhZv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/rBhZv.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-08-15T16:58:10.637Z
The area enclosed between the perpendiculars to the edges is also representative of these probabilities. This idea is generalizable to higher dimensions.
android16.5
CC BY-SA 4.0
2023-04-24T05:34:28.060Z
2023-04-24T11:53:05.863Z
2023-04-24T11:53:05.863Z
Amit
<p>A variable can take 3 values, each represented by a vertex of an equilateral triangle. Every point in the triangle is then a lottery over the vertices. How are the elements of each lottery related to its location in the triangle?</p>
<p><em>Hint:</em> The centroid, located at a distance of (1/√3,1/√3,1/√3) from each vertex, would represent the lottery (1/3,1/3,1/3).</p>

<uncertainty>
Lotteries in an equilateral triangle
android16.5
# Lotteries in an equilateral triangle

(Asked by: username_0 on 2023-04-21)

A variable can take 3 values, each represented by a vertex of an equilateral triangle. Every point in the triangle is then a lottery over the vertices. How are the elements of each lottery related to its location in the triangle?

*Hint:* The centroid, located at a distance of (1/√3,1/√3,1/√3) from each vertex, would represent the lottery (1/3,1/3,1/3).

**Tags:** uncertainty

---

## 1 answers

### Answer by username_1 on 2023-04-24. Score: 3 | Accepted answer

Consider an equilateral triangle of side length $\frac{2}{\sqrt{3}}$. If we pick any point inside the triangle and drop perpendiculars to the three sides of the triangle, then the length of these perpendiculars can represent the probabilities associated with three outcomes in a given lottery. See the graph below and verify that $p\_1+p\_2+p\_3=1$

[![enter image description here](https://i.sstatic.net/rBhZv.png)](https://i.sstatic.net/rBhZv.png)

> **Comments:**
> **username_0**: The area enclosed between the perpendiculars to the edges is also representative of these probabilities. This idea is generalizable to higher dimensions.  

---

