Expected Value After n Trials Suppose there is a game where a player can press a button for a chance to win a cash prize of $50,000. This can be done as many times as he/she wishes. The catch is that the chance of winning is 14/15 each time. A loss, which happens 1/15 times at random, will void all winnings and remove the player from the game. How many times should the player press the button to maximize their winnings?
This was a question brought up at dinner this evening by a family member with a slightly different game but same overall concept. It has been a while since I've done probability but I believe there is a solution to this problem. I assume that we can create a function using the expected value of each press and find the vertex to find the number of presses a player should do to maximize his/her prize. 
My guess is something like $y = ((14/15)^x*50,000) - ((1/15)^x*50,000x)$ where x is the number of trials, remembering that the player will potentially gain \$50,000 each round but also potentially lose all of their money (\$50,000x). 
Assuming there is a solution to this problem, is this the correct way to go about answering this? 
I apologize is this is completely off or a similar question has been asked, its been a few years since I've encountered a problem like this so I'm not even sure if I'm approaching/wording it the correct way.

Find n where the expression is maximized:
$$n(\frac{14}{15})^{n}$$
=> n = 14
