Winnings   Probability        Expectations
--------------------------------------------------
1900+18       1/757      1918(1/757)=  2.533685601
-18         756/757   (-18)(756/757)=-17.97622193
--------------------------------------------------
                  Total expectation =-15.44253633

If the same lottery were held many times and 757 tickets
were sold every time, and you bought 1 ticket every time,
then by the time you'd have played about 757 times, you'd 
have won about one time and you'd have averaged having 
lost about $15.44 per game.

Answer = -$15.44

Edwin

In such problems, usually two different expected values sit in and are considered for a gamer.


One expected value is the expected value of winning for a gamer.  It is 

     = 2.50990753 dollars.


In this specific problem, the winner pays $18 for his ticket, but then obtains 
these $18 back, so we can do not count these mutually annihilate operations.
Regarding the winner and the winning ticket, in this problem we only need to count the winning amount of $1900.



Another expected value is the expected value of the game for a gamer.  
In this problem, 756 tickets of 757 tickets lose $18 each, and one ticket of 757 tickets wins $1900.
So, the expected value of the game is 

       loosed     winning   final
       tickets    ticket    expectation

     +  = -15.4663144  dollars.



Since the problem does not concretize, which of the two possible expected values is of interest,
I make two conclusions from the post. 


    - First is that the problem, as it is worded, is posed incorrectly (it is incomplete).


    - Second one is that the author does not know a subject,
      does not know how the problem should be worded,
      and is not sure to which of the two expected values his question goes.