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You play a game with a coin. You may place a bet; if Heads is flipped then you receive your bet back plus the same in winnings. If Tails is flipped then you lose your bet. You have 20 dollars and you want to turn this into 40 dollars by continuously betting 1 dollar at a time, walking away when you either have a total of 40 dollars or are bankrupt. What is the probability you will leave with 40 dollars?

My logic and solution:

To get $40 I need to throw a coin at least 20 times as (40-20)/1=20. Using binomial distribution 20C20 (1/2)^20 * (1/2)^0 Is my answer correct? I know that most students solve it through a stochastic process, but I am not familiar with that course.

Thank you for your help in advance!

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    $\begingroup$ Think about the symmetry of 0 and 40 dollars wrt to the initial amount. $\endgroup$
    – gunes
    Commented Jun 11 at 20:19
  • $\begingroup$ This specific case has a simple answer due to the symmetry. In other cases you can either solve the distribution as a random walk (with a reflection principle applied) or easier, use a martingale approach with the optional stopping theorem. $\endgroup$
    – Sextus Empiricus
    Commented Jun 12 at 9:15
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    $\begingroup$ Related Probability that a simple 1d random walk is between [-k,k] in 100 moves $\endgroup$
    – Sextus Empiricus
    Commented Jun 12 at 9:19
  • $\begingroup$ In the case that the probability is not symmetric, or that the payout is not symmetric, then the approach with a martingale might work stats.stackexchange.com/a/401539 $\endgroup$
    – Sextus Empiricus
    Commented Jun 12 at 9:27
  • $\begingroup$ @Yuna can you tell something more about this problem. Where did you get it from, why do you try to solve this problem, what is your interest? $\endgroup$
    – Sextus Empiricus
    Commented Jun 13 at 14:17

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