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When to quit a coin toss doubling game?

$ \begingroup $ The game is as follows : I put in a dollar and if I get heads, I double my money. I can then continue playing and double my $ 2. Basically, I ‘m always allowed to continue playing and double the former sum. however, if it ‘s the coin lands on tails, I lose whatever come I ‘m presently playing for and have to restart the crippled ( which still would be a net loss of -1 because that ‘s what I paid to play ) .
Since I can stop the game at any point and cash my winnings out, when should I do that ?
Another assumption is that the casino has an infinite sum of money so it can play forever. however, although I ‘m very identical full-bodied and can play the game for a long fourth dimension, I ca n’t play it forever.

The start monetary value is $ 1 and the winnings double after each turn. A loss lone results in me losing the initial dollar and the potential of receiving more if I would ‘ve cashed out rather .
My question is whether there would be an “ optimum ” scheme play. That means, should I play the game and hope for, let ‘s say, 5 in a row, then cash out ( resulting in me receiving $ 32 ) and then start a newly game ? Or should I always cash out after 3 wins in a course ? possibly 10 wins ?
never cashing out is not an option since I ca n’t play the game constantly and at some period I would have no money left to play.

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At which point should I decide to collect my gain and then restart the game ? Will I finally go bankrupt or would I become infinitely rich at some point ?
EDIT : It is basically this question ( When to stop in this mint chuck game ? ) but the reward is not +100 but rather the double of the batch.

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EDIT 2 : I have thought more about this problem and it seems for me that the expected return should be zero. Let ‘s assume that on the third base cycle, I would win \ $ 8 ( 2 – > 4 – > 8 ). For that to happen, I would need to double my bet three times. The probability of that happening is $ \frac { 1 } { 8 } $, so in theory it should happen one out of eight games .
That would mean, that I would need to play 8 games and consequently pay a full of \ $ 8 to receive my winnings of \ $ 8. The lapp applies to $ 16 and every other sum .
Is that correct or am I missing something ?

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