*Magnicifecto* offers an “even value” game, with the catch that players may not leave the game if their (total) winnings are positive. (See previous question for more details.)

Scrooge has the gambling itch and decided that he could still afford to play, since he won in the first round. He recalled his brilliant strategy of doubling up each time, and decided to play the third round as follows:

He first makes a bet of $\$10$.

If he wins, he will **double** the size of his bet.

If his (total) winnings is ever non-positive, he will leave the game.

Now, what is the expected dollar amount of Scrooge's (total) winnings from this third round?