Work an At-Least-k Draw Probability from a Finite Pool Without Replacement — Free
Drawing three counters from a small jar changes what remains after each draw. Keep the finite population, marked count, draw count and at-least requirement together before using an independent-draw shortcut in a probability exercise.
The proof surface
Quartile explorers describe recorded datasets. This dial models an explicitly uniform finite-pool experiment, exposes combinatorial terms and handles impossible events. Alternative pools are compared separately rather than collapsed into a fictional combined probability.
InputPool code | total objects | marked objects | objects drawn; at least this many marked objects
Rare deviceQuartile explorers describe recorded datasets. This dial models an explicitly uniform finite-pool experiment, exposes combinatorial terms and handles impossible events. Alternative pools are compared separately rather than collapsed into a fictional combined probability.
Output artifactLargest at-least probability among alternative pools with row-by-row context
Cost$0 local calculation · no card, paid key, subscription or signup · proposed filing price is not for sale
Sample, not your facts: Illustrative inputs: Blue counters | 10 | 4 | 3; Pocket pack | 8 | 2 | 2. At least this many marked objects = 2. Largest at-least probability among alternative pools: 33.333333 % probability. All records are invented.
Before using the draw-probability dial
Assume every subset of the stated draw size is equally likely and objects are not replaced between draws. The marked classification must be fixed before drawing, and each physical object appears once in the pool. This is not a model for biased selection, changing populations or independent trials with replacement. Whole pool size is supported from one through five hundred, marked count from zero through pool size and draw count from zero through pool size. The requirement may exceed feasible successes, which yields a clear zero rather than an input crash. Log-factorial combination arithmetic keeps intermediate counts from overflowing, while the displayed result is a probability estimate rounded for presentation, not a simulation or a prediction of the next draw.
Why the flat version breaks
Using the marked fraction independently each time
After a without-replacement draw, pool composition changes. A binomial shortcut generally answers a different experiment. The hypergeometric count conditions on the finite pool and fixed draw size, which is why all three counts and the event requirement must remain visible.
Computing exactly k when the question says at least k
At least two includes two, three and any larger feasible count. Calculating only the two-success term omits favorable subsets. The dial lists the feasible successful-count range so the event wording is part of the arithmetic rather than an easily missed label.
Treating probability as a promise about the next draw
A one-third probability does not imply that every third attempt succeeds, and a past run of failures does not make success due. This tool calculates the declared experiment's probability; it does not read actual draws, detect bias or establish that a real selection process follows the assumption.
How to work the draw-probability dial
State the selection mechanism
Specify whether objects are returned after each draw and whether every eligible subset is equally likely. Without-replacement mathematics cannot be used just because a jar contains a small number of objects. If the drawing process is biased or changes the marked set, clarify that in the exercise rather than asking this calculator to infer probabilities from labels.
Enter the finite counts and event
Use a positive whole population, a whole marked count no greater than population and a whole draw count no greater than population. The setting is a non-negative whole at-least target. Neutral pool codes are sufficient. Different rows are alternative experiments; they need not share counts, but no automatic probability for doing all experiments is calculated.
Sum all feasible successful-count terms
For each success count j at least the target, multiply combinations of marked objects taken j at a time by combinations of unmarked objects taken draw-minus-j at a time. Divide by combinations of the full population taken draw at a time. Sum only feasible j. The reading shows the feasible range and the event, with explicit zero or certainty conventions at boundaries.
Check the assumptions with the exercise source
Compare a small pool by enumerating its possible subsets or use the worked combinatorial counts. Ask an instructor to distinguish at least from exactly and with replacement from without replacement. A rounded probability says nothing about whether an observed outcome is guaranteed, suspicious or due on the next draw.
What the draw-probability dial separates
Question
Before
Inspect this instead
Using the marked fraction independently each time
After a without-replacement draw, pool composition changes. A binomial shortcut generally answers a different experiment. The hypergeometric count conditions on the finite pool and fixed draw size, which is why all three counts and the event requirement must remain visible.
Enter the finite counts and event
Computing exactly k when the question says at least k
At least two includes two, three and any larger feasible count. Calculating only the two-success term omits favorable subsets. The dial lists the feasible successful-count range so the event wording is part of the arithmetic rather than an easily missed label.
Sum all feasible successful-count terms
Treating probability as a promise about the next draw
A one-third probability does not imply that every third attempt succeeds, and a past run of failures does not make success due. This tool calculates the declared experiment's probability; it does not read actual draws, detect bias or establish that a real selection process follows the assumption.
Check the assumptions with the exercise source
This draw-probability dial replaces a manual count or calculation, not source verification or the responsible person’s review.
Run it on the samples, right here
FIRST-LOAD
HYPOTHESIS / PROTOTYPE — checkout unavailable. Calculation is local. A draft is saved automatically in this browser profile when storage is available; Reset to sample clears it. Optional Pro history stores only five summaries and has its own deletion control. State links encode your inputs and can remain in browser history, clipboard or recipients’ records; share only non-sensitive rows. Optional external AI formatting leaves this device. The required site analytics beacon reports page activity; shared URLs contain encoded inputs. Do not treat an encoded URL as private. The calculator has no input-collection endpoint.
Finite-pool probability exercise only, not a prediction, gambling strategy or proof that an observed selection was fair. Confirm the sampling mechanism and event wording with an instructor or qualified analyst before applying the model to real observations.
Data note: The draw-probability dial processes Pool code | total objects | marked objects | objects drawn locally. Starter/sample selection and Run compute in this tab; no input is sent by the calculator. A local draft may be saved; explicit state-link sharing or optional external AI formatting can disclose inputs. Use non-sensitive labels.
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Finite-pool probability exercise only, not a prediction, gambling strategy or proof that an observed selection was fair. Confirm the sampling mechanism and event wording with an instructor or qualified analyst before applying the model to real observations.
What this is built on
Method: For each success count j at least the target, multiply combinations of marked objects taken j at a time by combinations of unmarked objects taken draw-minus-j at a time. Divide by combinations of the full population taken draw at a time. Sum only feasible j. The reading shows the feasible range and the event, with explicit zero or certainty conventions at boundaries.
All sample records, dates, quantities and labels are invented. No outside policy, contract, rate, clock offset, measurement or accessibility standard is represented as verified.
Google’s official pricing documentation, fetched 2026-10-01, says AI Studio is free in available regions. Optional formatting may require a Google account; manual local entry requires none. Limits can change and free-tier content may be used to improve products. Do not send private records.
Before: repeated draws were treated as independent by habit. After: finite composition and the full at-least event have explicit combinatorial workings.
Setting: At least this many marked objects = 2. Expected summary: 33.333333 % probability.
For Blue, exactly two marked draws have C(4,2)×C(6,1)=36 combinations, and exactly three have C(4,3)×C(6,0)=4. Divide forty favorable combinations by C(10,3)=120 to obtain one third, or 33.333333%. Pocket needs both marked objects in two draws: C(2,2)/C(8,2)=1/28, about 3.571429%. The summary reports the largest alternative-pool probability; it does not add independent experiments.
Sample B — changed plan
Tray A | 12 | 5 | 4
Tray B | 6 | 3 | 2
Setting: At least this many marked objects = 2. Expected summary: 57.575758 % probability.
Tray A's favorable counts for two, three and four marked objects are 210, 70 and 5, totaling 285 out of C(12,4)=495. Its probability is 57.575758%. Tray B has three favorable two-marked pairs out of fifteen possible pairs, or 20%. The comparison keeps draw sizes and pool composition explicit rather than using the marked proportion as though draws were independent with replacement.
Sample C — boundary convention
Nearly unmarked jar | 7 | 1 | 2
Setting: At least this many marked objects = 2. Expected summary: 0 % probability.
The requirement is at least two marked objects, but the jar contains only one. The event is impossible under this finite-pool model and its probability is zero, with a review note naming the constraint. Drawing no objects would likewise make any positive target impossible. For an explicitly zero target, at least zero is certain; that is a different entered event, not an error.
Optional AI formatting, never the calculation
Manual entry completes this draw-probability dial for free without signup. If available to you, the free AI Studio interface linked in the sources may format fictional or non-sensitive notes; external access may require an account. No API key or AI call is built into this tool. Free-tier content may be used to improve products. Review each cell and transcribe it to the labeled row schema; do not paste the JSON object into the row box.
Format only these fictional or non-sensitive notes for a draw-probability dial. Return strict JSON shaped as {"rows": [{"label": "string", "cells": ["string", "string", "string"]}], "setting": "string"}. The columns are Pool code | total objects | marked objects | objects drawn; the setting is At least this many marked objects. Keep all supplied strings and quantities exactly; do not calculate, infer missing entries, invent dates or add advice. If any required value is missing, return an empty rows array and ask me for it separately. I will verify every cell against my source and manually transcribe rows using vertical bars before running the local calculator.
An AI response is not executed, fetched or trusted as a result. Missing values remain questions; the strict local parser checks the rows you actually enter.