SCHWEITZER 1993 Problem 10

Let $U_1 , U_2 , U_3$ be iid random variables on [0,1], which in order of magnitude, $U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}$ . Let $\alpha, p_1 , p_2 , p_3 \in [0,1]$ such that $P(U_j ^ {\ast} \ge p_j)= \alpha$ ( j = 1,2,3). Prove that $$P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha$$