Gambler's ruin special case

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dikshaarvindsinh

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A casino with infinite monetary resources invites Anirudh to spend a day on the condition that he only carries $1 with him. Anirudh in turn, insists that the casino plays fair on any particular round. The casino honours his request, and they agree on the following rules. On each toss of a fair coin Anirudh gets $1 from the casino if it is a Head, and Anirudh must give $1 to the casino if it is a tail. The game ends if Anirudh has no money left. What is the probability that Anirudh will walk away with nothing? Also, what should the probability p of Anirudh’s winning be (on any given toss), in order to ensure an even probability of him going bankrupt?
 
dikshaarvindsinh said:
A casino with infinite monetary resources invites Anirudh to spend a day on the condition that he only carries $1 with him. Anirudh in turn, insists that the casino plays fair on any particular round. The casino honours his request, and they agree on the following rules. On each toss of a fair coin Anirudh gets $1 from the casino if it is a Head, and Anirudh must give $1 to the casino if it is a tail. The game ends if Anirudh has no money left. What is the probability that Anirudh will walk away with nothing? Also, what should the probability p of Anirudh’s winning be (on any given toss), in order to ensure an even probability of him going bankrupt?
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I tried solving this through random walk and got the answer 2/3 by infinity gp and for the second part i think it'll be 1/2 because they are asking about the possibility of winning at any given coin. I am pretty sure thats not how I am supoosed to do it but I am afraid that I havent been taught the tools required to solve this question.
 
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