Gambling With Rich Friends

Yudhisthira is a habitual gambler and gambles at every opportunity he can find. One day, he enters in a casino with 1 rupee in his pocket and starts betting. At each bet, he is likely to win 1 rupee with probability \(0.1\) and likely to lose 1 rupee with probability \(0.9\). Yudhisthira also has a super-supportive rich friend, Krishna, who always provides with 1 rupee to keep him betting whenever Yudhisthira's total earning becomes zero.

What is the probability that Yudhisthira will eventually earn 1 million rupees from the gambling?


Note: If \(q_t\) denotes the amount of money that Yudhisthira's has after \(t\) bets, we have \[ q_{t+1}= \max(1,q_t+G_t),\] where \(G_t\) is an independent random variable where \(G_t=+1 \) with probability \(0.1\) and \(G_t=-1\) with probability \(0.9\).

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