The bold strategy part is what makes this annoying.
Model the problem as a Markov chain with states representing token counts, and set up equations for the probability of reaching 5 before 0. Solve the equations step by step, starting from the state closest to the target, and compute the final probability for state 3.
Pro tip: Always verify your solution by checking boundary conditions and ensuring probabilities are between 0 and 1; also, consider if the betting strategy is optimal or if there's a simpler way to see the answer.
Identify the possible token counts (0,1,2,3,4,5) and determine the betting amount from each state based on the rule: bet the maximum that won't overshoot 5.
Let P(n) be the probability of reaching 5 before 0 starting with n tokens. Write equations using the win/loss probabilities and the transitions from each state.
Use the known values P(0)=0 and P(5)=1, and solve the equations backwards from state 4 down to state 1.
Substitute the solved values into the equation for state 3 to find the final probability.
Check that the probability is between 0 and 1, and briefly discuss the result in the context of the problem.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.