Find the Success Probability of Bold Betting
Company: Sig
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Online Assessment
# 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.
### What a Strong Answer Covers
- The exact next state from each fortune 1 through 4.
- Boundary probabilities at fortunes 0 and 5.
- A recurrence using the `1/3` win and `2/3` loss probabilities.
- A solved, checked value for the starting state 2.
### 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?
Overview: 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.
Read the full Sig Data Scientist interview experience this question came from