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Use Bayes for factory identification

Last updated: Mar 29, 2026

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.

SIG (Susquehanna)

Aug 11, 2025, 12:00 AM

Data Scientist

Take-home Project

Statistics & Math

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).

Solution

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