{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-prob-foundations",
    "area": "mathematics",
    "topic": "probability",
    "origin": "original",
    "title": "Counting, total probability, and independence",
    "skills": [
      "counting",
      "conditional-probability",
      "independence"
    ],
    "priority": "foundation",
    "difficulty": "easy",
    "estimated_minutes": 7,
    "prerequisites": [
      "fractions"
    ],
    "prompt": "A class has $6$ students, $4$ of whom use Python. Two students are selected uniformly without replacement. Find the probability that both use Python. Then state whether the two selection indicators are independent.\n",
    "answer": "The probability is\n$$\n\\frac{\\binom{4}{2}}{\\binom{6}{2}} = \\frac{6}{15} = \\frac{2}{5}.\n$$\nThe indicators are not independent because sampling without replacement changes the second probability after the first outcome.\n",
    "check": {
      "kind": "numeric",
      "id": "original-prob-foundations",
      "values": [
        0.4
      ]
    }
  }
}