{
  "id": "2308.14635",
  "version": "v1",
  "published": "2023-08-28T15:04:24.000Z",
  "updated": "2023-08-28T15:04:24.000Z",
  "title": "Expected Number of Dice Rolls Until an Increasing Run of Three",
  "authors": [
    "Daniel Chen"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "A closed form is found for the expected number of rolls of a fair n-sided die until three consecutive increasing values are seen. The answer is rational, and the greatest common divisor of the numerator and denominator is given in terms of n. As n goes to infinity, the probability generating function is found for the limiting case, which is also the exponential generating function for permutations ending in a double rise and without other double rises. Thus exact values are found for the limiting expectation and variance, which are approximately 7.92437 and 27.98133 respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-28T15:04:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "expected number",
      "dice rolls",
      "increasing run",
      "double rise",
      "greatest common divisor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}