I stared at this for a second because the 'permanent rider' part felt open-ended.
Clarify the assumptions about the experiment: each ride is offered independently with probability P, and the rider's decision to take a ride is also independent with probability P. Then model the number of rides taken on the first day as a Binomial(2, P) distribution, and compute the probability of exactly one ride. For the probability of becoming a permanent rider, recognize that this requires taking both rides on the first day (probability P^2), and then assume that if they take both, they will continue riding every day thereafter.
Pro tip: Always state your assumptions explicitly and note that in reality, rider behavior is likely correlated (e.g., if they like the first ride, they may be more likely to take the second). This shows you understand the limitations of the model and can think critically about experiment design.
Confirm that each ride is offered independently with probability P, and that the rider's decision to take each ride is also independent with probability P. Define X as the number of rides taken on the first day.
Since there are two rides, X follows a Binomial(2, P) distribution. The probability of exactly one ride is P(X=1) = 2P(1-P).
A permanent rider is defined as someone who rides every day going forward. This requires taking both rides on the first day (X=2), with probability P^2. Assume that if they take both, they become permanent with probability 1.
The probability of exactly one ride is 2P(1-P). The probability of becoming a permanent rider is P^2.
Mention that this is a simplified model and that in practice, probabilities may not be independent, and the definition of 'permanent' may require additional conditions.
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