Start by clarifying that the question is underspecified and requires assumptions about how day-one experience affects future ride behavior. Propose a simple probabilistic model, such as a Markov chain, where the probability of taking a ride the next day depends on whether the rider had a good or bad experience on day one. Then compute the requested probabilities under that model, and discuss how you would validate or refine the model with real data.
Pro tip: Explicitly state your assumptions and note that in a real analysis you would estimate these transition probabilities from experimental data rather than assume them. This shows you understand the difference between a theoretical exercise and practical data science.
Identify that the question lacks details on how day-one experience influences future behavior. Assume each ride independently has probability P of being priced reasonably, and define how this affects the rider's likelihood to take future rides.
Propose a model where the rider's probability of taking a ride on day two depends on the number of reasonably priced rides on day one. For example, let q_k be the probability of taking a ride on day two given k good experiences on day one.
Calculate the probability of having 0, 1, or 2 reasonably priced rides on day one using the binomial distribution with parameters n=2 and probability P.
Use the law of total probability: sum over k=0,1,2 of P(k good rides on day one) * P(take exactly one ride on day two | k good rides). If the rider takes at most one ride per day, then this is simply the probability they take a ride on day two.
Define 'permanent daily rider' as taking a ride every day from day two onward. If the daily decision is independent given the day-one experience, this probability is the product of daily ride probabilities from day two onward, which may be zero unless the daily probability is 1. Alternatively, model it as an absorbing state in a Markov chain.
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