# Read binomial probability in Book Isle: check repeated independence before exactly two successes
Language: en
Canonical: [Read binomial probability in Book Isle: check repeated independence before exactly two successes]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T09:59:34+08:00
Author: Bison
List six success-position patterns in a four-draw replacement model, separating exactly two from at least two successes.
English translation of the [Chinese original](), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.
The probability of two successes in four trials is not one success probability times one failure probability. Read original draw cards in Book Isle, fixing independence, count and constant probability before counting success positions. Use a probability model with explicit assumptions, not private-data sampling or automatic independence testing and prediction.
## State the experiment and success definition
Use an original model with one red and two blue balls: draw uniformly, record, replace and remix under identical conditions four times. Red means success and blue failure, purely counting labels. Record source and replacement rule before using formulas. This is a paper model, not experimental verification of randomness. Real processes need independent checks rather than inheriting these ideal assumptions.
## Match conditions to parameters
There are four fixed trials, two outcome categories and independent identical success probability one third; failure probability is two thirds. Record n, p and failure probability, with X counting zero through four successes. A blue draw does not make red “due” next. Independence is a probability assumption, not merely separate clicks or records; reader navigation cannot make dependent data independent.
## Calculate one specified pattern
For the specified pattern `SSFF`, multiply one-third squared by two-thirds squared to obtain four eighty-firsts. Four factors cover all trials, including failures. Preserve pattern and probability; this is not yet the total for exactly two successes. Changed success probability or replacement conditions require new factors rather than a permanent four-eighty-firsts answer for every two-red/two-blue problem.
## List all six success-position patterns
Exactly two successes have patterns `SSFF`, `SFSF`, `SFFS`, `FSSF`, `FSFS`, `FFSS`, six disjoint possibilities. Each has probability four eighty-firsts under the assumptions, giving twenty-four eighty-firsts or eight twenty-sevenths. Choosing two of four positions independently confirms six. Do not multiply again for distinguishable red labels: only success positions are counted. Retain both the enumeration and what the coefficient counts.
## Add three and four for at least two
At least two also includes exactly three and four. Exactly three has four positions totalling eight eighty-firsts; exactly four has one eighty-first. Total thirty-three eighty-firsts equals eleven twenty-sevenths, not eight twenty-sevenths. Label events and mark “exactly” versus “at least”. The number of trials remains four: the wording selects X values rather than how many draws occur. Typography does not change that scope.
## Check a no-replacement control before calculating
Without replacement, a first red leaves two blues and next success probability zero; a first blue leaves one red/one blue and next probability one half. The original three-ball bag also cannot support four draws without replacement. Rewrite feasible rules before applying a model. Do not invent an extra ball to preserve the formula or blame the reader: conditions changed. Keep original and control separate.
## Check missing patterns and mean-versus-outcome errors
Four eighty-firsts suggests only SSFF was counted; twelve patterns suggests duplicate position labels. Expected count four thirds does not mean a fractional ball in one run or a guaranteed result; actual X is an integer zero through four. Missing trial count stays unknown rather than inferred from four display lines. Share original rules/patterns and external notes. Unperformed sampling remains unperformed, and reading statistics are not probability-trial data.
## Reopen with rules, patterns and events
Reopen, state replacement/four trials, list six patterns and recalculate eight twenty-sevenths versus at-least eleven twenty-sevenths. Retain factors, disjoint addition, parameters and no-replacement counterexample, not just a combination coefficient. Reassess changed probability, count or process; bookmarks locate sources without preserving model validity. Deliver reproducible conditional reasoning that separates assumptions, calculations and unperformed observations rather than claim real prediction from one correct example.
## Related guides and sources
[Read sampling without replacement with Book Isle: update the remaining balls at every level](); [Book Isle counting problems: distinguish selecting two people from assigning two roles]().
Checked on November 15, 2026: [Apple public information]() lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the [official App Store]() for published capabilities; verify controls in the installed version.
Method reference: [OpenStax: binomial-experiment conditions and distribution](). Materials and numbers are original exercises. Record external reports, paper reasoning and actual reading observations separately, without inventing missing conditions. Sources do not establish new automatic analysis by the app. No step-by-step physical-device, private-server or all-output-environment testing is claimed.