Fair price per door with two prizes in seven

A game show has $7$ closed doors; $2$ of them hide a prize of $\text{\char36}200$ and the remaining $5$ are empty, with no clue as to which is which. You open doors one at a time at random, never reopening one, and stop as soon as you uncover a prize. If each door you open costs $\text{\char36}x$, what value of $x$ makes the game fair?

$75$ dollars. The two prize doors split the five empty ones into three gaps of equal expected size $\tfrac53$, so the expected number of doors opened is $\tfrac53+1=\tfrac83$. Fairness requires $\tfrac83x=200$, giving $x=75$.