{
  "index": "2002-B-1",
  "type": "COMB",
  "tag": [
    "COMB",
    "NT"
  ],
  "difficulty": "",
  "question": "Shanille O'Keal shoots free throws on a basketball court. She hits\nthe first and misses the second, and thereafter the probability that\nshe hits the next shot is equal to the proportion of shots she\nhas hit so far. What is the probability she hits exactly 50 of\nher first 100 shots?",
  "solution": "The probability is $1/99$. In fact, we show by induction on $n$\nthat after $n$ shots, the probability of having made any number of\nshots from $1$ to $n-1$ is equal to $1/(n-1)$. This is evident\nfor $n=2$. Given the result for $n$, we see that the probability of\nmaking $i$ shots after $n+1$ attempts is\n\\begin{align*}\n\\frac{i-1}{n} \\frac{1}{n-1} + \\left( 1 - \\frac{i}{n} \\right) \\frac{1}{n-1}\n&= \\frac{(i-1) + (n-i)}{n(n-1)} \\\\\n&= \\frac{1}{n},\n\\end{align*}\nas claimed.",
  "vars": [
    "n",
    "i"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "shotcount",
        "i": "hitscount"
      },
      "question": "Shanille O'Keal shoots free throws on a basketball court. She hits\nthe first and misses the second, and thereafter the probability that\nshe hits the next shot is equal to the proportion of shots she\nhas hit so far. What is the probability she hits exactly 50 of\nher first 100 shots?",
      "solution": "The probability is $1/99$. In fact, we show by induction on $shotcount$\nthat after $shotcount$ shots, the probability of having made any number of\nshots from $1$ to $shotcount-1$ is equal to $1/(shotcount-1)$. This is evident\nfor $shotcount=2$. Given the result for $shotcount$, we see that the probability of\nmaking $hitscount$ shots after $shotcount+1$ attempts is\n\\begin{align*}\n\\frac{hitscount-1}{shotcount} \\frac{1}{shotcount-1} + \\left( 1 - \\frac{hitscount}{shotcount} \\right) \\frac{1}{shotcount-1}\n&= \\frac{(hitscount-1) + (shotcount-hitscount)}{shotcount(shotcount-1)} \\\\\n&= \\frac{1}{shotcount},\n\\end{align*}\nas claimed."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "pineapple",
        "i": "chandelier"
      },
      "question": "Shanille O'Keal shoots free throws on a basketball court. She hits\nthe first and misses the second, and thereafter the probability that\nshe hits the next shot is equal to the proportion of shots she\nhas hit so far. What is the probability she hits exactly 50 of\nher first 100 shots?",
      "solution": "The probability is $1/99$. In fact, we show by induction on $pineapple$\nthat after $pineapple$ shots, the probability of having made any number of\nshots from $1$ to $pineapple-1$ is equal to $1/(pineapple-1)$. This is evident\nfor $pineapple=2$. Given the result for $pineapple$, we see that the probability of\nmaking $chandelier$ shots after $pineapple+1$ attempts is\n\\begin{align*}\n\\frac{chandelier-1}{pineapple} \\frac{1}{pineapple-1} + \\left( 1 - \\frac{chandelier}{pineapple} \\right) \\frac{1}{pineapple-1}\n&= \\frac{(chandelier-1) + (pineapple-chandelier)}{pineapple(pineapple-1)} \\\\\n&= \\frac{1}{pineapple},\n\\end{align*}\nas claimed."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "emptiness",
        "i": "totality"
      },
      "question": "Shanille O'Keal shoots free throws on a basketball court. She hits\nthe first and misses the second, and thereafter the probability that\nshe hits the next shot is equal to the proportion of shots she\nhas hit so far. What is the probability she hits exactly 50 of\nher first 100 shots?",
      "solution": "The probability is $1/99$. In fact, we show by induction on $emptiness$\nthat after $emptiness$ shots, the probability of having made any number of\nshots from $1$ to $emptiness-1$ is equal to $1/(emptiness-1)$. This is evident\nfor $emptiness=2$. Given the result for $emptiness$, we see that the probability of\nmaking $totality$ shots after $emptiness+1$ attempts is\n\\begin{align*}\n\\frac{totality-1}{emptiness} \\frac{1}{emptiness-1} + \\left( 1 - \\frac{totality}{emptiness} \\right) \\frac{1}{emptiness-1}\n&= \\frac{(totality-1) + (emptiness-totality)}{emptiness(emptiness-1)} \\\\\n&= \\frac{1}{emptiness},\n\\end{align*}\nas claimed."
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "i": "hjgrksla"
      },
      "question": "Shanille O'Keal shoots free throws on a basketball court. She hits\nthe first and misses the second, and thereafter the probability that\nshe hits the next shot is equal to the proportion of shots she\nhas hit so far. What is the probability she hits exactly 50 of\nher first 100 shots?",
      "solution": "The probability is $1/99$. In fact, we show by induction on $qzxwvtnp$\nthat after $qzxwvtnp$ shots, the probability of having made any number of\nshots from $1$ to $qzxwvtnp-1$ is equal to $1/(qzxwvtnp-1)$. This is evident\nfor $qzxwvtnp=2$. Given the result for $qzxwvtnp$, we see that the probability of\nmaking $hjgrksla$ shots after $qzxwvtnp+1$ attempts is\n\\begin{align*}\n\\frac{hjgrksla-1}{qzxwvtnp} \\frac{1}{qzxwvtnp-1} + \\left( 1 - \\frac{hjgrksla}{qzxwvtnp} \\right) \\frac{1}{qzxwvtnp-1}\n&= \\frac{(hjgrksla-1) + (qzxwvtnp-hjgrksla)}{qzxwvtnp(qzxwvtnp-1)} \\\\\n&= \\frac{1}{qzxwvtnp},\n\\end{align*}\nas claimed."
    },
    "kernel_variant": {
      "question": "Artemis Quill is practising penalty kicks on the soccer pitch. She \nmisses her first kick and scores her second, and thereafter the probability \nthat she scores the next kick is always equal to the proportion of kicks she \nhas scored so far. What is the probability that she scores exactly $32$ of \nher first $80$ kicks?",
      "solution": "Let P_n(i) denote the probability that after n≥2 kicks Artemis has scored exactly i of them.\n\nBase case n=2. By hypothesis she has taken one miss and one score, so the only possible hit–count is 1. Hence P_2(1)=1.\n\nInductive hypothesis. Assume that for some fixed n≥2 the distribution {P_n(i):1≤i≤n−1} is uniform, i.e.\nP_n(i)=1/(n−1) for 1≤i≤n−1.\n\nTransition to n+1. Conditionally on having i successes after n kicks,\n• the probability the next kick is a success equals the current proportion of successes, i/n;\n• the probability it is a miss equals 1−i/n.\n\nTherefore for 1≤i≤n,\nP_{n+1}(i)\n   = (i−1)/n·P_n(i−1) + (1−i/n)·P_n(i).\nSubstituting the inductive hypothesis gives\nP_{n+1}(i)\n  = (i−1)/(n(n−1)) + (n−i)/(n(n−1))\n  = 1/n.\nThus the distribution on {1,…,n} is uniform with mass 1/n each, completing the induction.\n\nApplying the result with n=80, the probability of any specific hit–count between 1 and 79 is 1/79.  In particular,\nPr{exactly 32 successes in the first 80 kicks}=1/79.",
      "_meta": {
        "core_steps": [
          "Base case n=2: after two predetermined shots (one hit, one miss) the only possible hit-count is 1, hence the distribution on {1} is trivially uniform.",
          "Inductive hypothesis: for some n≥2 the hit-count distribution on {1,…,n−1} is uniform with probability 1/(n−1) each.",
          "Transition relation: P_{n+1}(i)=((i−1)/n)·P_n(i−1)+(1−i/n)·P_n(i) obtained from the rule ‘next-shot success probability = current hit proportion’.",
          "Algebraic simplification shows P_{n+1}(i)=1/n for 1≤i≤n, completing the induction and maintaining uniformity.",
          "Apply the uniformity at n=100 to get Prob{50 hits}=1/99."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Name of the shooter.",
            "original": "Shanille O'Keal"
          },
          "slot2": {
            "description": "Context/sport phrasing of the attempts.",
            "original": "free throws on a basketball court"
          },
          "slot3": {
            "description": "Fixed outcomes of the first two shots (must be exactly one hit and one miss, order arbitrary).",
            "original": "hits the first and misses the second"
          },
          "slot4": {
            "description": "Total number of shots under consideration.",
            "original": 100
          },
          "slot5": {
            "description": "Required number of hits among those shots (must lie between 1 and total−1).",
            "original": 50
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}