Fair entry fee per door when hunting a prize

On a game show, one of $7$ closed doors hides $200 and the rest are empty; you have no idea which. You open doors one at a time, each choice random among the unopened ones, until you find the money, and each door you open costs you $x$ dollars. What value of $x$ makes this a fair game?

$x=50$. The position of the prize in your random opening order is uniform on $1,\dots,7$, so you open $4$ doors on average (not geometric, since you never reopen a door). Fairness needs $4x=200$, i.e. $x=50$ dollars.