Estimate free-trial conversion probability

Context

A product ships with a free trial. You are asked: "For a group of people who began a free trial, what is the probability they convert?"

Data (assume available)

A table of trial starts:

• user_id
• trial_start_at (timestamp)
• trial_length_days (int)
• country/channel/plan/device (optional segments)

A table of paid conversions:

• user_id
• paid_start_at (timestamp, null when the user never converted)
• revenue (optional)

Task

1. Spell out exactly what "conversion probability" refers to, covering the time window and the cohort definition.
2. Show how you would estimate the conversion probability for a cohort of trial starters, including how you would deal with:
   • users whose trial is still running (incomplete observation)
   • conversions that arrive late (payment begins after the trial)
   • segmentation and possible confounding (e.g., a shift in channel mix)
3. Give a way to quantify uncertainty (confidence or credible intervals) and to communicate the results.
4. Propose follow-up analyses/metrics (diagnostics and guardrails) that keep the estimate useful for decisions.

Overview: This question tests a data scientist's competency in cohort definition, censored-data handling, segmentation and confounding assessment, and uncertainty quantification when measuring free-trial to paid conversion, and it belongs to the Analytics & Experimentation domain within data science.

Read the complete IBM Data Scientist interview write-up this question is drawn from