Gambler’s Fallacy - Round 2

Probability Level 4

The Las Vegas Casino Magnicifecto had some success attracting their hotel guests to play their “even value” game. Whatever bet size the player places (say $A \$A), there is a 50% chance that he will get +$A +\$A , and a 50% chance that he will get $A - \$A . However, they quickly realized that they were losing money and decided to modify the rules drastically: players may not leave the game if their (total) winnings are positive.

Scrooge, who was on vacation, was eager to continue playing this game after his previous success. However, he was slightly confused about the new ruling, and decided to play it safe. He plays the second round of the game as follows:

He first makes a bet of exactly $10 \$10.
If his (total) winnings are not positive, he will leave the game.
Otherwise, he will continue to make a bet of exactly $10 \$10 .

Now, what is the expected value of Scrooge’s (total) winnings from this second round?

Image credit: Wikipedia Fluteflute

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