# Bayes rule with a base rate

> ML Prep screening item.

- Stable ID: `math-prob-02-bayes-base-rate`
- Collection: screening
- Area: Mathematics
- Topic: Probability
- Origin: original
- Difficulty: medium
- Expected time: 4 minutes
- Skills: conditional-probability, total-probability, bayes-rule, probabilistic-independence
- Item JSON: https://mlprep.iwase.dev/item/math-prob-02-bayes-base-rate.json

## Problem

A condition has prevalence $P(D)=0.10$. A test has sensitivity $P(+\mid D)=0.80$
and false-positive rate $P(+\mid D^c)=0.20$. Find $P(+)$ and $P(D\mid +)$.
Are the events $D$ and $+$ independent?


## Answer policy

The human page contains a collapsed answer. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-prob-02-bayes-base-rate`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.