{ "category": "Statistical functions", "display_name": "PROB", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.PROB", "implementation_state": { "answer_bank": null, "detail": "Recorded: 8 workbook cells.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "matches", "state_label": "Matches every recorded Excel answer", "summary": "Matches all 8 recorded answers", "workbook_corpus": { "cells": 8, "matched": 8 } }, "implementations": [ { "commit": "9e1884369a678158dd82d3d74188daf986995e87", "flavour": "excel-compatibility-candidate", "language": "Rust", "name": "CalcFunc", "own_code": [ "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L412-L418", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L504-L517", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/lookup_prob_frequency_family.rs#L16-L46" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L412-L418", "state": "matches", "suite": null, "summary": "Matches all 8 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.PROB.json", "name": "PROB", "notes": [], "schema": "efh.function-reference/v1", "section_order": [ "what-it-does", "try-it", "quirks", "how-to-implement", "version-differences", "alternatives" ], "sections": { "alternatives": "\n### Relationships\n\n- **[FREQUENCY](FUNC.FREQUENCY.md)** is `PROB`'s empirical twin. `FREQUENCY` counts observations\n falling into bins; `PROB` sums a *given* probability vector over an interval. Feed\n `FREQUENCY`'s counts through a division by the sample size and you have a `prob_range` that\n `PROB` will accept.\n- **[BINOM.DIST](FUNC.BINOM.DIST.md), [POISSON.DIST](FUNC.POISSON.DIST.md),\n `HYPGEOM.DIST`** are the named discrete distributions. Their cumulative\n forms answer the same question `PROB` answers, but from a parameterized family rather than from\n a tabulated one. `PROB` is what you use when the distribution came from data or from a model\n Excel does not name.\n- **`SUMIFS`** computes the same sum with the interval expressed as criteria:\n `SUMIFS(prob, x, \">=\"&a, x, \"<=\"&b)`. What `SUMIFS` will not do is check that `prob` is a\n probability vector — which is `PROB`'s entire added value and the source of both of its\n `#NUM!` conditions.\n- Readers confuse `PROB` with `PERCENTILE`/`QUARTILE`, which invert an *empirical* distribution\n built from raw observations rather than reading a supplied probability vector.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\n`PROB` looks arithmetically trivial and is not, for one reason: it contains an equality test\nagainst `1` in disguise.\n\nThe accumulation itself is a plain left-to-right sum of at most `n` doubles, each in `[0, 1]`.\nThere is no cancellation — every term is non-negative — so the relative error of the returned\nmass is bounded by roughly `n` units in the last place of the running total, which is the best\nbehaviour a naive sum ever has. Compensated summation (Kahan, or the doubly compensated variants\nanalysed in Higham, *Accuracy and Stability of Numerical Algorithms*, chapter 4) would tighten\nthat to a constant, and would be worth having only for very long support vectors.\n\nThe validation sum is the delicate part. `sum(p_i)` is computed the same naive way and then\ncompared against `1`. Three separate things now depend on floating-point detail:\n\n1. **Whether the input is representable at all.** Probability vectors that are exact in decimal\n are usually not exact in binary64. Ten copies of `0.1` do not sum to `1`; neither do three\n copies of `1/3`. A function that tested equality literally would reject most hand-written\n distributions, which is why a tolerance exists.\n3. **The tolerance width itself.** It is a free parameter with observable consequences at the\n boundary. Widening it admits vectors that are not distributions; narrowing it rejects vectors\n that are. There is no principled value, only a chosen one, and Microsoft's documentation\n supplies none.\n\nThe accumulated interval mass and the validation sum are also computed independently, so `PROB`\nover the whole support is not guaranteed to return exactly `1` even when the validation passed.\nAn implementation that wanted that identity would have to return the validation sum itself when\nthe interval covers every atom, which no source suggests Excel does.\n\n\n", "quirks": "\n### Result and edge cases\n\n\nReturns `Number` in `[0, 1]`.\n\n- **No atom falls in the interval.** The sum over an empty index set is `0`. `PROB` returns zero,\n not an error and not `#N/A`. A caller cannot distinguish \"the interval genuinely has zero mass\"\n from \"the interval missed every atom\" — they are the same statement about the distribution.\n- **Errors in the vectors** propagate as themselves.\n\n\n", "sources": "- Microsoft, *PROB function* —\n \n (signature, the omitted-`upper_limit` point-probability rule, the `#N/A` unequal-count\n condition, and the two `#NUM!` conditions on `prob_range`). Retrieval was blocked by the\n upstream host for this page; the documented behaviour above is stated as documented behaviour\n and should be re-checked against the page.\n- Handbook `data/functions/FUNC.PROB.json`, `data/presence/FUNC.PROB.json` (arity, classification\n axes, implementing module and its sibling surfaces).\n- Handbook [coercion and lifting](../model/02-coercion-and-lifting.md) — the direct-argument\n versus range-scan split that `PROB`'s strict collection sits outside of.\n- N. J. Higham, *Accuracy and Stability of Numerical Algorithms*, 2nd ed., chapter 4 (summation\n error bounds and compensated summation) — the analysis behind the summation-order remark.\n\n", "version-differences": "No comparison between specific Excel builds is recorded here yet. This does not establish that all builds behave alike.", "what-it-does": "\n`PROB(x_range, prob_range, lower_limit, [upper_limit])` treats its two vector arguments as a\n**finite discrete probability distribution** and returns the mass that distribution assigns to a\nclosed interval.\n\nWrite the paired vectors as `(x_1, p_1), …, (x_n, p_n)`. They define the discrete measure\n\n mu = sum_{i=1..n} p_i * delta_{x_i}\n\non the real line, where `delta_x` is the unit point mass at `x`. `PROB` returns\n\n PROB(x, p, a, b) = mu([a, b]) = sum over { i : a <= x_i <= b } of p_i\n\nTwo features of that definition carry the page:\n\n1. **The interval is closed at both ends.** The accumulation admits `x_i` when\n `a <= x_i` and `x_i <= b`. So `PROB` computes `P(a <= X <= b)`, not `P(a < X <= b)` and not\n `P(a <= X < b)`. On a discrete distribution the difference is a whole atom, not a\n measure-zero technicality: it is exactly `p` at the endpoint.\n\n\nThe range of the function is `[0, 1]`, and it is monotone in the interval: widening `[a, b]`\nnever decreases the result. `PROB` is a probability *lookup*, not a distribution fit — it does\nno smoothing, no interpolation between the `x_i`, and no normalization. A value of `lower_limit`\nthat lies strictly between two atoms contributes nothing.\n\n`x_range` is not required to be sorted, and the atoms are not required to be distinct. If the\nsame `x` appears twice with different probabilities, both are admitted and both are counted;\n`PROB` sums over index positions, not over distinct support points.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Default |\n|---|---|---|\n| `x_range` | The support: the numeric values `X` can take. Required. | — |\n| `prob_range` | The probabilities attached to those values, positionally. Required. | — |\n| `lower_limit` | The lower bound of the interval, inclusive. Required. | — |\n| `upper_limit` | The upper bound, inclusive. Optional. | `lower_limit` |\n\nThe pairing is **positional**, element by element in row-major order over each argument. That is\nthe whole of the correspondence: `x_range` and `prob_range` are two columns of one table, and\nnothing checks that they were meant to line up beyond their having the same number of elements.\n\n`prob_range` is documented to be a genuine probability vector: each entry in `(0, 1]`, and the\nentries summing to 1. Both conditions are stated by Microsoft as `#NUM!` conditions rather than\nas advice, which makes `PROB` unusually strict for a worksheet function — most of Excel will\ncompute happily on nonsense.\n\n`lower_limit` and `upper_limit` are ordinary numeric slots subject to\n[coercion and lifting](../model/02-coercion-and-lifting.md).\n\n\n" }, "signature": { "display": "PROB(x_range, prob_range, lower_limit, [upper_limit])", "parameters": [ { "description": null, "name": "x_range", "optional": false, "repeats": false }, { "description": null, "name": "prob_range", "optional": false, "repeats": false }, { "description": null, "name": "lower_limit", "optional": false, "repeats": false }, { "description": null, "name": "upper_limit", "optional": true, "repeats": false } ], "trailing_repeats": false }, "sources": { "metadata": { "admission": { "label": "implemented", "reason": null }, "arity": { "max": 4, "min": 3 }, "category": "Statistical functions", "classification": { "arg_preparation_profile": "RefsVisibleInAdapter", "coercion_lift_profile": "Custom", "determinism_class": "Deterministic", "error_collapse_profile": "None", "fec_dependency_profile": "RefOnly", "host_interaction_class": "None", "kernel_signature_class": "Custom", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "RefOnly", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Probability that values in a range fall between two limits, from their probabilities", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the probability that values in a range are between two limits", "microsoft_url": "https://support.microsoft.com/en-us/office/prob-function-9ac30561-c81c-4259-8253-34f0a238fc49" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "PROB" }, { "locale": "bg-BG", "name": "PROB" }, { "locale": "cs-CZ", "name": "PROB" }, { "locale": "da-DK", "name": "SANDSYNLIGHED" }, { "locale": "de-DE", "name": "PROB" }, { "locale": "el-GR", "name": "PROB" }, { "locale": "en-AU", "name": "PROB" }, { "locale": "en-GB", "name": "PROB" }, { "locale": "en-US", "name": "PROB" }, { "locale": "es-ES", "name": "PROBABILIDAD" }, { "locale": "et-EE", "name": "PROB" }, { "locale": "fi-FI", "name": "TODENNÄKÖISYYS" }, { "locale": "fr-FR", "name": "PROBABILITE*" }, { "locale": "he-IL", "name": "PROB" }, { "locale": "hr-HR", "name": "PROB" }, { "locale": "hu-HU", "name": "VALÓSZÍNŰSÉG" }, { "locale": "id-ID", "name": "PROB" }, { "locale": "it-IT", "name": "PROB" }, { "locale": "ja-JP", "name": "PROB" }, { "locale": "ko-KR", "name": "PROB" }, { "locale": "lt-LT", "name": "PROB" }, { "locale": "lv-LV", "name": "PROB" }, { "locale": "nb-NO", "name": "PROB" }, { "locale": "nl-NL", "name": "KANS" }, { "locale": "pl-PL", "name": "PRAWDPD" }, { "locale": "pt-BR", "name": "PROB" }, { "locale": "pt-PT", "name": "PROB" }, { "locale": "ro-RO", "name": "PROB" }, { "locale": "ru-RU", "name": "ВЕРОЯТНОСТЬ" }, { "locale": "sk-SK", "name": "PROB" }, { "locale": "sl-SI", "name": "PROB" }, { "locale": "sr-Latn-RS", "name": "PROB" }, { "locale": "sv-SE", "name": "SANNOLIKHET" }, { "locale": "th-TH", "name": "PROB" }, { "locale": "tr-TR", "name": "OLASILIK işlevi" }, { "locale": "uk-UA", "name": "PROB" }, { "locale": "vi-VN", "name": "PROB" }, { "locale": "zh-CN", "name": "PROB" }, { "locale": "zh-HK", "name": "PROB" }, { "locale": "zh-TW", "name": "PROB" } ], "signature": { "display": "PROB(x_range, prob_range, lower_limit, [upper_limit])", "parameters": [ { "description": null, "name": "x_range", "optional": false, "repeats": false }, { "description": null, "name": "prob_range", "optional": false, "repeats": false }, { "description": null, "name": "lower_limit", "optional": false, "repeats": false }, { "description": null, "name": "upper_limit", "optional": true, "repeats": false } ], "trailing_repeats": false }, "surface_name": "PROB", "version_marker": null, "xlcall": { "code": "317", "symbol": "xlfProb" } }, "microsoft": "https://support.microsoft.com/en-us/office/prob-function-9ac30561-c81c-4259-8253-34f0a238fc49" }, "try_it": { "formula": "=PROB({1;2;3},{0.2;0.3;0.5},2,3)" }, "url": "https://handbook.dnacalc.com/functions/PROB.html", "variants": [], "versions": [] }