{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $p$ and $q$ be natural numbers with $p \u003e q$.\nIn an election, candidate $A$ gets $p$ votes and candidate $B$ gets $q$ votes.\nThe votes are counted one at a time, in an order chosen uniformly at random among all orders.\nThen the probability that $A$ is strictly ahead of $B$ after each vote is counted is\n$$\\frac{p - q}{p + q}.$$\n\n## Remarks\n\nEquivalently, the number of such counting orders, as words with $p$ letters $A$ and $q$ letters $B$, is $\\frac{p - q}{p + q} \\binom{p + q}{p}$.\nExample: for $p = 3$ and $q = 1$, the probability is $\\frac{1}{2}$. Of the $4$ orders, $AAAB$ and $AABA$ work.\nA short proof uses the reflection principle for [[lattice paths|040WY0TT2G]].\n\n## History\n\nWilliam Allen Whitworth published the result in 1878.\nJoseph Bertrand found it again in 1887, and the theorem has his name.\n\n## Explanations\n\n### L1 human\n\nIf the winner gets many more votes than the loser, the winner is likely to lead the whole time.\nThe formula makes this precise: the chance of an unbroken lead is the margin divided by the total number of votes.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "05A15"
    ],
    "name": "Bertrand's ballot theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2253746",
      "wikipedia": "Bertrand's ballot theorem"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Bertrand's_ballot_theorem"
      }
    ],
    "tag": "0FS972T0ZH",
    "uses": [
      "092GEF38YD"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0FS972T0ZH/",
  "links": [
    {
      "field": "uses",
      "target": "092GEF38YD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0FS972T0ZH.md",
  "msc": [
    "05A15"
  ],
  "name": "Bertrand's ballot theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0FS/0FS972T0ZH.md",
  "status": null,
  "tag": "0FS972T0ZH",
  "uri": "m2://0FS972T0ZH",
  "url": "https://math2.tamnd.com/wiki/bertrands-ballot-theorem/",
  "writing": {
    "tag": "0FS972T0ZH",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}