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Question 27229: An Exact Binomial Lattice Is Not a Normal Approximation

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0053-27229 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0054

Original question and givens

Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).

  • G001 — Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [clt-binomial-lattice]

After

derived = [[n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope]]

Conditions: none

List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

CLT binomial ORIGINAL one minus probability := sub(1, 1/2)

After

CLT binomial ORIGINAL one minus probability = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [CLT binomial k0 success, 1/2, 0]

After

derived = [1]

Conditions: none

A zeroth power is the empty product1,not0 because n or the base is0.

Step identifiers and dependencies

Rule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S002

S004 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

CLT binomial ORIGINAL power product k0 := mul(1, 1/8)

After

CLT binomial ORIGINAL power product k0 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

CLT binomial ORIGINAL choose times joint k0 := mul(1, 1/8)

After

CLT binomial ORIGINAL choose times joint k0 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [3, 0, 1/2, 1/2]

After

derived = [1, 1, 1/8, 1/8]

Conditions: none

Every choice of k success positions has probability p^k(1-p)^(n-k),with C(n,k) choices.This complete finite-independence calculation gives an exact lattice mass,not a normal approximation.Later sums reuse the SAME established pure term recipe without changing any source profile.

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S008

S010 · calculation

Before

CLT binomial k1 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k1 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

CLT binomial ORIGINAL power product k1 := mul(1/2, 1/4)

After

CLT binomial ORIGINAL power product k1 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

CLT binomial ORIGINAL choose times joint k1 := mul(3, 1/8)

After

CLT binomial ORIGINAL choose times joint k1 = 3/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

source roles = [3, 1, 1/2, 1/2]

After

derived = [3, 1/2, 1/4, 3/8]

Conditions: none

Every choice of k success positions has probability p^k(1-p)^(n-k),with C(n,k) choices.This complete finite-independence calculation gives an exact lattice mass,not a normal approximation.Later sums reuse the SAME established pure term recipe without changing any source profile.

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S014

S016 · calculation

Before

CLT binomial k2 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k2 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

CLT binomial k2 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k2 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

CLT binomial k2 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k2 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

CLT binomial ORIGINAL power product k2 := mul(1/4, 1/2)

After

CLT binomial ORIGINAL power product k2 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

CLT binomial ORIGINAL choose times joint k2 := mul(3, 1/8)

After

CLT binomial ORIGINAL choose times joint k2 = 3/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [3, 2, 1/2, 1/2]

After

derived = [3, 1/4, 1/2, 3/8]

Conditions: none

Every choice of k success positions has probability p^k(1-p)^(n-k),with C(n,k) choices.This complete finite-independence calculation gives an exact lattice mass,not a normal approximation.Later sums reuse the SAME established pure term recipe without changing any source profile.

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S020

S022 · calculation

Before

CLT binomial k3 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k3 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

CLT binomial k3 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k3 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

CLT binomial k3 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k3 success ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

source roles = [CLT binomial k3 failure, 1/2, 0]

After

derived = [1]

Conditions: none

A zeroth power is the empty product1,not0 because n or the base is0.

Step identifiers and dependencies

Rule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S024

S026 · calculation

Before

CLT binomial ORIGINAL power product k3 := mul(1/8, 1)

After

CLT binomial ORIGINAL power product k3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

CLT binomial ORIGINAL choose times joint k3 := mul(1, 1/8)

After

CLT binomial ORIGINAL choose times joint k3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [3, 3, 1/2, 1/2]

After

derived = [1, 1/8, 1, 1/8]

Conditions: none

Every choice of k success positions has probability p^k(1-p)^(n-k),with C(n,k) choices.This complete finite-independence calculation gives an exact lattice mass,not a normal approximation.Later sums reuse the SAME established pure term recipe without changing any source profile.

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S027

S029 · calculation

Before

CLT binomial ORIGINAL total mass sum0 := sum(0, 1/8)

After

CLT binomial ORIGINAL total mass sum0 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum0 := sum(0, 1/8)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum0 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

CLT binomial ORIGINAL total mass sum1 := sum(1/8, 3/8)

After

CLT binomial ORIGINAL total mass sum1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

CLT binomial ORIGINAL total mass sum2 := sum(1/2, 3/8)

After

CLT binomial ORIGINAL total mass sum2 = 7/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

CLT binomial ORIGINAL total mass sum3 := sum(7/8, 1/8)

After

CLT binomial ORIGINAL total mass sum3 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [3, 1/2, 0, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]]]

After

derived = [1, 1/8, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), No]

Conditions: none

List every k=0..n;the total is(p+(1-p))^n=1.The CDF includes k<=threshold,equality included.The z expression is a symbolic standardized threshold,not a Gaussian approximation,error bound or general CLT proof.

Step identifiers and dependencies

Rule: CLT_complete_binomial_total_inclusive_CDF_NO_normal_approximation · Uses: G001, S033

Reverse checks

  • S034 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S033 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum3 with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum2 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum1 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum0 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum0 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S027 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k3 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k3 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S024 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S020 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k2 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k2 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S014 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute CLT binomial k1 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S008 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S002 · premise_and_result_audit — Recompute CLT binomial ORIGINAL one minus probability with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 4e10c1b59e2de0f76cfde794e996d2f6608934406007ac879dc019837ac24cb2
  • candidate record SHA-256: 23acc52ff19fd67a355747157c356dc144d7650f40017a2990f90e118f916517
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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