 The probability triangle (aka “Marschak-Machina triangle”) offers a helpful visual representation of preferences over the space of lotteries over \(\{x_1, x_2, x_3\}\), with \(x_3\succ x_2 \succ x_1\). Since, for any \(P\in \mathcal{P}\), \(P(x_2)= 1- P(x_1)-P(x_3)\), one can represent the situation two-dimensionally, with lotteries appearing as points in a unit triangle in which the horizontal axis gives us \(P(x_1)\) and the vertical one gives us \(P(x_3)\). The northwestern, southwestern and southeastern corners respectively correspond to the lotteries yielding \(x_3, x_2\) and \(x_1\) for sure.