You are playing a game where there are 9 boxes laid out in a row, numbered 1 through 9 from left to right. Randomly placed in one of the boxes is a slip of paper that says "GAME OVER". The other eight boxes each contain $1000.
You are to pick the boxes one at a time. If you pick a box with $1000, you keep the money and you must pick another box. If at any point you select the box that says "GAME OVER", the game ends and you leave with the prize money you've accumulated to that point. The only catch is, the host of the game show will tell you which direction the "GAME OVER" box is in, and you must guess your next box in that direction (it's like a guessing game where you have to guess a number from 1 to 9, and after each guess the host tells you "higher" or "lower" until you finally guess the number he is thinking of). However, the goal of this game is not to land on the "GAME OVER" box (since you eventually will), but to maximize the number of guesses you take (and thus your profit) before you land on it.
Question 1: Is there an optimal strategy for this game? If so, what is it and what is your expected profit? If not, why not?
Question 2: What if you were the host, and instead of randomly placing the "GAME OVER" box, you could choose where it went - is there a strategy that would minimize the expected profit of the contestant?
there is no optimal strategy because the "GAME OVER" box is placed at random and therefore there is no strategy that will ensure you to get $8,000 because you could do the strategy one way but the next time the first box you pick could be the "GAME OVER" the only thing remotely to a strategy would be to try to find where the game over box is by picking either box 4 or 6 then when the host pointed somewhere picking the last box or the first box depending on which way he pointed and then you know what numbers the "GAME OVER" lies between, unless you pick the "GAME OVER" box in either your first or second try. Which is why there is a feeble strategy but the game is mostly played on luck.
Question2. The best position to place the "GAME OVER" box would be either far left or far right, therefore forcing the person to eventually be forced to select the GAME OVER" unless the person chooses the boxes starting in the opposite corner and going to the box adjacent to the other box