If a person were to bet on a fair coin toss and on each toss placed a dollar bet on heads with the strategy to quit once ahead by a dollar or after 20 tosses (whichever comes sooner). Assume an infinite bank roll. What is the probability of this person winning $1?
Tossing a fair coin with intention to quit once ahead
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$\begingroup$
probability
probability-theory
gambling
2 Answers
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HINT: This is the same as the probability that the first passage time of a simple random walk to $S_n =1$ is less than or equal to $20.$ The first passage time $\tau(1)$ has distribution $$P(\tau(1) = 2m-1) = \frac{P(S_{2m-1}=1)}{2m-1}.$$
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Found this which answers similar question.