Find the Success Probability of Bold Betting
SIG (Susquehanna)Data Scientist · Online Assessment
Last updated
Calculate the chance of reaching five tokens before ruin under a bold, state-dependent betting rule. The solution enumerates the transitions, sets absorbing boundaries, solves the four-state recurrence for win probability one-third, and obtains 15/77 from an initial fortune of two.
Find the Success Probability of Bold Betting
You begin with 2 tokens and want to reach 5 before reaching 0. On each turn, you bet as many tokens as possible without betting more than needed to reach 5: at fortune s, the stake is min(s, 5 - s). You win each bet independently with probability 1/3; a win adds the stake and a loss subtracts it.
What is the probability of reaching 5 tokens before 0? Express the answer as a fraction in simplest form.
Follow-up questions
- How would the recurrence change for an arbitrary win probability `p`?
- Why does this bold strategy visit only a small set of states from the initial fortune?
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