Gamblers Always Lose - Round 4
The Las Vegas Casino
Magnicifecto
offers an “even value” game, with the catch that players may not leave the game players may not leave the game if their (total) winnings are positive. See
previous question
for more details.)
Scrooge wasn’t too happy that the previous game didn’t go too well. He decided that he wasn’t daring enough, and so decided to triple up, instead of just doubling up. His strategy is as follows:
He first makes a bet of . If he wins, he will triple 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 fourth round?
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1 solution
take nth game:
probability of him getting out of the round in nth game = (1/2)^n he must have won all the past n-1 games, past profit = (10+10 3+10 3^2+...10 3^(n-2)) = 10 (3^(n-1) - 1) /2 if he does gets out of the round by loosing in the nth game, loss in nth game = 10 3^(n-1) total score = 10 (3^(n-1) - 1) /2 - 10*3^(n-1) = -5 (3^(n-1)+1 )
Expected score = sigma p.x = sigma (1/2)^n * -5 (3^(n-1) +1) = sigma (-5/3)(3/2)^n + sigma -5 (1/2)^n The first term is geometric series with ratio = 3/2 >1 hence it will diverge to -infinity, so expected score = -infinity