Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between $\tfrac{1}{2}$ and $\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?
I understand the correct solution, but I don’t know why my solution is wrong.
My solution:
The probability Alice chooses between 0 and 1/2 is 1/2. The number carol should choose here is 1/2. The probability Alice chooses between 1/2 and 2/3 is 1/6. The number carol should choose here is 7/12. The probability Alice chooses between 2/3 and 1 is 1/3. The number carol should choose here is 2/3.
Therefore carol should choose $1/2*1/2+1/6*7/12*1/3*2/3=41/72$
But it’s not the correct answer, can someone explain why?
Best Answer
Your answer doesn't work because the optimal strategy is not necessarily the probability-weighted mean of the optimal strategies under different conditions. For instance, the probability that Carol wins in the first and third cases (where Alice chooses a number in between $0$ and $\frac{1}{2}$; and where Alice chooses a number between $\frac{2}{3}$ and $1$), is $1$, but the average of these two strategies does not yield a choice that wins in all cases in which Alice chooses a number in the union of the two disjoint intervals because the win probability now depends on what Bob chooses. Therefore, your fundamental approach to the question is incorrect. Instead, you must choose a number $c \in [0,1]$ for Carol, and compute the unconditional probability that $c$ lies between $a$ and $b$ for $a \in [0,1]$ and $b \in [1/2, 2/3]$ as a function of $c$; then find the $c$ that maximizes this function.