Use Bayes for factory identification

Quick Overview

This question evaluates understanding of Bayesian inference and conditional probability by asking for the posterior probability that two observed black widgets originated from a specific factory.

Use Bayes for factory identification

Company: SIG (Susquehanna)

Role: Data Scientist

Category: Statistics & Math

Difficulty: easy

Interview Round: Online Assessment

Factory A produces 40% red and 60% black widgets; Factory B produces 80% red and 20% black widgets. One factory is chosen uniformly at random, and then two widgets are sampled uniformly at random from that factory. Given that both sampled widgets are black, what is the probability they came from Factory A?

Overview: This question evaluates understanding of Bayesian inference and conditional probability by asking for the posterior probability that two observed black widgets originated from a specific factory.

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SIG (Susquehanna)
Aug 11, 2025
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Bayesian inference: Which factory given two black widgets?

Setup

  • Two factories produce red and black widgets.
    • Factory A: 40% red, 60% black.
    • Factory B: 80% red, 20% black.
  • One factory is chosen uniformly at random (each with probability 1/2).
  • Two widgets are then sampled uniformly at random from that chosen factory.
  • You observe that both sampled widgets are black.

Assumption: draws are independent conditional on the factory (equivalently, sampling with replacement or from a very large batch).

Task

Compute the posterior probability that the widgets came from Factory A given that both sampled widgets are black: P(A | both black).

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