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The game rules are: bet an amount, flip a fair coin, if heads I win the bet and get the amount, if tails I lose the bet and pay the amount. I quit after getting my first head. If I lose, I double the bet and play again. What are my expected earnings (start with a $1 bet)?

I visualized a few rounds and saw that I can play until I get net \$1 back. So I always can regain my losses. So I think expected earnings is $1$? I'm not sure how to justify this answer.

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    $\begingroup$ "I can play until I get net $1 back". That is only absolutely true if you have a potentially unlimited bankroll, a potentially unlimited amount of time and an opponent who is prepared to accept potentially unlimited large bets. In the real world where there are limits, with a fair coin and fair odds, your expected earnings remain zero throughout $\endgroup$ – Henry Oct 8 '16 at 21:04
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HINT: You have a Geometric distribution here. This has a pmf as $f(x) = (1-p)^x\cdot p$ where $p$ is the probability of getting heads. Can you find the expected value of this and apply it to the betting game?

Edit: the geometric distribution has two schools of thought. the pmf that I gave you comes from counting the number of failures before one success. The other idea, which I think you are using, is counting the number of trials before one success. the pmf for this is $f(x) = (1-p)^{x-1}p$

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  • $\begingroup$ So expected value of success = [1-p(success)]/p(success) = 1, since p(success) = 1/2 $\endgroup$ – MoronicHero Oct 8 '16 at 19:37
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Let me show it this way:

Scenario #1 - You win

Good job! you made money and can go home with your winnings

Scenario # 2 - You lose

you are a dollar down, so you bet two dollars, if you win then you lose $1, then won a two dollars, net profit is 1 dollar, you can go home now

Conclusion

this is called the Martingale system, and is primarily used for blackjack, but has the same principal: if your original stake is one, then you keep doubling and your profit will always be 1 (unless there is a maximum bet rule) because in the series 1 + 2 + 4 + 8 + 16... the sum of all the previous terms PLUS ONE equals the next term, and that is why you should always profit

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  • $\begingroup$ ok but is there a way to show the pmf of this? $\endgroup$ – MoronicHero Oct 8 '16 at 17:50
  • $\begingroup$ @MoronicHero show that for any given term in the series 1 + 2 + 4 + 8... that all the previous terms added together will never be more than the next term, or specifically, always one less, or that if your term is N, then the previous is N/2, then before that is N/4 and befor ethat is N/8... which is a sequence we all know aproaches one, but never reaches it $\endgroup$ – Cursed Oct 8 '16 at 17:52

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