# Read dependence from a joint table

> ML Prep screening item.

- Stable ID: `math-prob-05-joint-correlation`
- Collection: screening
- Area: Mathematics
- Topic: Probability
- Origin: original
- Difficulty: hard
- Expected time: 6 minutes
- Skills: joint-distributions, marginal-distributions, conditional-distributions, covariance, correlation
- Item JSON: https://mlprep.iwase.dev/item/math-prob-05-joint-correlation.json

## Problem

Binary variables $X,Y\in\{0,1\}$ have joint probabilities
$$
p_{00}=0.3,\quad p_{01}=0.2,\quad p_{10}=0.1,\quad p_{11}=0.4
$$
Find $P(X=1)$, $P(Y=1)$, $P(X=1\mid Y=1)$, $\operatorname{Cov}(X,Y)$,
and $\operatorname{Corr}(X,Y)$.


## 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-05-joint-correlation`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.