# Condition first, then remove conditioning

> ML Prep screening item.

- Stable ID: `math-prob-06-total-variance`
- Collection: screening
- Area: Mathematics
- Topic: Probability
- Origin: original
- Difficulty: medium
- Expected time: 4 minutes
- Skills: conditional-expectation, law-total-expectation, law-total-variance
- Item JSON: https://mlprep.iwase.dev/item/math-prob-06-total-variance.json

## Problem

Let $Z\sim\operatorname{Bernoulli}(1/4)$. Conditional on $Z=0$, $X$ has mean $0$
and variance $1$; conditional on $Z=1$, $X$ has mean $4$ and variance $1$.
Find $E[X\mid Z]$, $E[X]$, and $\operatorname{Var}(X)$.


## 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-06-total-variance`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.