Infer the Factory from Two Black Widgets
Company: Sig
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Online Assessment
# Infer the Factory from Two Black Widgets
Factory A produces 40% red widgets and 60% black widgets. Factory B produces 80% red widgets and 20% black widgets. A factory is selected uniformly at random, and then two widgets are sampled independently from that factory's stable color distribution. Both sampled widgets are black.
What is the probability that Factory B was selected? Express the answer as a fraction in simplest form.
### What a Strong Answer Covers
- Equal prior probabilities for the two factories.
- The two-black likelihood under each factory.
- Bayes' rule with a normalized denominator.
- The role of the stated independent-sampling assumption.
### Follow-up Questions
- How would the answer change after observing one black and one red widget?
- What changes if sampling occurs without replacement from known finite inventories?
Overview: Use Bayes' rule to identify a factory after observing two black widgets. The solution combines equal priors with factory-specific two-draw likelihoods, derives a posterior probability of 1/10 for Factory B, and explains why the independent-sampling assumption matters.
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