Lyft · Probability & Brainteasers

Commuter Coupon Conditional Probability

TrueInterview

August 21, 2026 · 1 min read

Requirements

The data science statistics screen begins with a handful of SQL and pandas warm‑ups, then presents a conditional‑probability question set in Lyft’s couponing work:

Context: After rolling out targeted coupons successfully, the team now wants to test whether coupons can nudge riders toward commuting with Lyft. Picture N commuters who each make two trips a day — one to work and one back home. For each trip the chance they take Lyft is:

Probability of choosing Lyft to work = p, with p < 1/2

Probability of choosing Lyft for the return home = 2p if they came to work via Lyft; otherwise it stays p

Question 1: What is the probability a rider goes home with Lyft? Question 2: Given a rider took Lyft home, what is the probability they also took Lyft to work?

Solve Question 1 by applying the law of total probability over the to‑work choice.
Solve Question 2 with Bayes’ rule, feeding the answer from Question 1 into the denominator.
Name your assumptions clearly and explain why the bound p < 1/2 keeps 2p a legitimate probability.

Notes

This is the opening technical filter for the data science loop; the SQL and pandas exercises come first, then this probability question.
Derive the marginal before the posterior: merge the took‑Lyft‑to‑work branch (return probability 2p) and the did‑not‑take‑Lyft‑to‑work branch (return probability p), then condition with Bayes’ rule for the second part.
Keep the algebra symbolic in p rather than plugging in a number early — the interviewer is evaluating the total‑probability and Bayes structure, not a numeric shortcut.

Preparation

Derive both solutions manually in terms of p, then spot‑check at a concrete value like p = 0.3.
Drill the law of total probability and Bayes’ rule on two‑phase conditional setups until the marginal‑then‑posterior flow becomes second nature.
Warm up the statistics screen’s first half independently: rehearse common aggregation, window‑function, and groupby/merge tasks in SQL and pandas under a clock.

Requirements

The data science statistics screen begins with a handful of SQL and pandas warm‑ups, then presents a conditional‑probability question set in Lyft’s couponing work:

Context: After rolling out targeted coupons successfully, the team now wants to test whether coupons can nudge riders toward commuting with Lyft. Picture N commuters who each make two trips a day — one to work and one back home. For each trip the chance they take Lyft is:

Probability of choosing Lyft to work = p, with p < 1/2
Probability of choosing Lyft for the return home = 2p if they came to work via Lyft; otherwise it stays p

Question 1: What is the probability a rider goes home with Lyft? Question 2: Given a rider took Lyft home, what is the probability they also took Lyft to work?

Solve Question 1 by applying the law of total probability over the to‑work choice.

Solve Question 2 with Bayes’ rule, feeding the answer from Question 1 into the denominator.

Name your assumptions clearly and explain why the bound p < 1/2 keeps 2p a legitimate probability.

Notes

This is the opening technical filter for the data science loop; the SQL and pandas exercises come first, then this probability question.

Derive the marginal before the posterior: merge the took‑Lyft‑to‑work branch (return probability 2p) and the did‑not‑take‑Lyft‑to‑work branch (return probability p), then condition with Bayes’ rule for the second part.

Keep the algebra symbolic in p rather than plugging in a number early — the interviewer is evaluating the total‑probability and Bayes structure, not a numeric shortcut.

Preparation

Derive both solutions manually in terms of p, then spot‑check at a concrete value like p = 0.3.

Drill the law of total probability and Bayes’ rule on two‑phase conditional setups until the marginal‑then‑posterior flow becomes second nature.

Warm up the statistics screen’s first half independently: rehearse common aggregation, window‑function, and groupby/merge tasks in SQL and pandas under a clock.