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Imagine you have $k$ dollars in your pocket and you are gambling with a wealthy man (with infinitely much money). The rule is repeatedly tossing a coin and you win $\$1$ if it's a head, otherwise you lose $\$1$. Now, what's the probability that you get broke eventually?

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Gambler's Ruin. – Ragib Zaman Oct 6 '12 at 7:18
The probability is $1$. – Rasmus Oct 6 '12 at 7:19

This is a very common question in stochastic processes known as Gambler's ruin problem.

the answer is :If coin is fair,gambler will go broke with probability $1$.

Also it should be noted that if the coin is biased,i.e, probability of gambler winning is greater that $0.5$, then there is strictly positive probabilty that Gambler is never broke.

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Which is 1-((1-p)/p)^g where g is Gambler's initial fortune and p>.5 is the probability that Gambler wins a single toss. – Did Oct 6 '12 at 8:29

The probability is $1$. This is the gambler’s ruin problem, and see also one-dimensional random walks.

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