Biased Coin Game Win Probability

This biased coin betting game is a classic Markov chain question in quant prep, framed as a simple wealth exchange between two players. The dynamics form a tiny finite-state stochastic process with clear absorbing states, making it ideal to see how Markov chains apply to gambling-style interview questions without heavy notation. It turns a casual game into a structured probabilistic model, exactly the kind of modeling mindset quant interviews probe.

It trains recognition of Markov structure, state definition, absorbing probabilities, and recursive conditioning. You practice translating a story into states and transitions, writing down the right equations, using boundary conditions, and checking that the resulting probabilities are sensible. This tightens intuition for discrete-time stochastic processes that underlie more complex models.

It matters because front-office quant interviews, grad-scheme screens, and hedge fund quant prep frequently use such coin-toss and gambler's ruin variants to test comfort with Markov chains, risk of ruin reasoning, and probabilistic thinking under bias. Understanding this style of problem helps with questions on default risk, barrier options, and other path-dependent yet Markovian models that appear in real quantitative finance interviews.

What it tests

This problem class is governed by the principle of Markov processes and recursive conditioning: when a process has a finite set of states and transitions probabilistically between them, the probability of reaching a particular absorbing state can be expressed in terms of the probabilities from neighboring states. The key is that the future evolution depends only on the current state, not the path taken to reach it (the Markov property). By considering the possible immediate transitions and their probabilities, you can set up equations relating the probability of winning from each state. Solving these equations, often by leveraging boundary conditions (such as one player having all the money), yields the desired probabilities. This approach works because the process is memoryless and the transitions are well-defined and finite.