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Compute Commuter Ride Probabilities

LyftData Scientist · Technical Screen
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This question tests the ability to apply Bayes' Theorem and marginal probability to model commuter behavior and predict ride-hailing demand based on environmental factors.

Scenario

At a major transit hub, commuters decide between taking a Lyft or using public transit. Historical data indicates that 60% of commuters prefer Lyft. On days with heavy rain, 80% of those who prefer Lyft will request a ride, while only 20% of public transit users will switch to Lyft. On clear days, 30% of Lyft-preferring commuters request a ride, and only 5% of public transit users switch to Lyft. Assume that the probability of a rainy day is 25%.

Task

  1. Calculate the total probability that a randomly selected commuter will request a Lyft on a given day.
  2. If a commuter requests a Lyft, what is the probability that it is currently raining?
  3. Let X be an indicator variable where X=1 if a commuter requests a Lyft and X=0 otherwise. Calculate the expected value E[X] for a randomly selected commuter.

Constraints

  • Assume P(Rain) = 0.25, P(Clear) = 0.75.
  • P(Lyft_Pref) = 0.60, P(Transit_Pref) = 0.40.
  • Provide your answers as decimals rounded to four places.

Example

If 100 people pass through the hub, and the probability of a commuter requesting a Lyft is P(L), the expected number of rides is 100 * P(L). Show the step-by-step breakdown using the Law of Total Probability.

Hints

Hint 1

Define the events clearly: R for Rain, C for Clear, L for Lyft Request, and M for Lyft-preferring commuter.

Hint 2

Use the Law of Total Probability to find P(L) by partitioning the space into (Rain, Lyft-Pref), (Rain, Transit-Pref), (Clear, Lyft-Pref), and (Clear, Transit-Pref).

Hint 3

Apply Bayes' Theorem: P(R|L) = (P(L|R) * P(R)) / P(L), where P(L|R) is the probability of a request given it is raining.

Follow-up questions

  • How would the calculation change if we assume the probability of rain is not independent of the number of commuters present?
  • If the cost of a Lyft ride increases during rain, how might that introduce a selection bias in our observed data?
  • Suppose we want to predict ride volume for the next 24 hours; what other variables besides weather should we incorporate to improve our model's precision?