Professor $X$ chooses a number $x$ between $a$ and $b$ ($a < b \in \mathbb{Z}$) and asks two other students $A_1$ and $A_2$ to pick numbers between $a$ and $b$. Whichever is closer to $x$ gets $90\%$ and the other $80\%$, if $a$ and $b$ are distinct. Otherwise, they both get $85\%$. If $X$'s and $A_1$'s numbers are chosen at random and $A_1$ announces his number out loud, describe a strategy that leads $A_2$ to the highest possible mark.
I'm not sure how $A_2$ can get the highest possible mark. A suitable sample space for this problem would be $\Omega = \{(a_1,a_2,a_3) : a_i\in \{a,\cdots, b\}\,\forall i\},$ where $a_1$ is the Professor's choice, $a_2$ is $A_1$'s choice, and $a_3$ is $A_2$'s choice. There is a $\frac{b-a}{2(b-a+1)}$ probability that $A_1$ chooses a number below the Professor's, a $\frac{1}{b-a+1}$ probability that $A_1$ chooses a number equal to the Professors, and a $\frac{b-a}{2(b-a+1)}$ probability that $A_1$ chooses a number greater than the Professor's, but I'm not sure if this is useful. One can calculate these probabilities using the law of total probability and, defining $A =\{(a_1,a_2,a_3) : a_i \in \{a,\cdots, b\}, a_2 < a_1\},$ by defining $B_i := \{(i,a_1,a_2) : a_j \in \{a,\cdots, b\}, 1\leq j\leq 2\}$ for $a\leq i\leq b$, and then computing $\sum_{i=1}^{b-a+1} P(A | B_i)P(B_i).$ However, I'm not sure if this is useful for solving the problem. I think $A_2$ should choose either $\lfloor \dfrac{a+a_2}{b-a+1}\rfloor $ or $\lfloor \dfrac{b+a_2}{b-a+1}\rfloor$ (i.e. the number in between the lowest possible choice and $A_1$'s choice or in between the highest possible choice and $A_1$'s choice), though this is just from my intuition.
