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Question 27242: 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

R0053-27242 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0067

Original question and givens

Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=4, 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=5, integer threshold=4, 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 k0 failure ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply4 = 1/16

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 k0 failure ORIGINAL power multiply5 := mul(1/16, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply5 = 1/32

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 · calculation

Before

CLT binomial ORIGINAL power product k0 := mul(1, 1/32)

After

CLT binomial ORIGINAL power product k0 = 1/32

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, S008

S010 · calculation

Before

CLT binomial ORIGINAL choose times joint k0 := mul(1, 1/32)

After

CLT binomial ORIGINAL choose times joint k0 = 1/32

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 · implies

Before

source roles = [5, 0, 1/2, 1/2]

After

derived = [1, 1, 1/32, 1/32]

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, S010

S012 · 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, S011

S013 · 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, S012

S014 · 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, S013

S015 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k1 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, S014

S016 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply4 = 1/16

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 ORIGINAL power product k1 := mul(1/2, 1/16)

After

CLT binomial ORIGINAL power product k1 = 1/32

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 ORIGINAL choose times joint k1 := mul(5, 1/32)

After

CLT binomial ORIGINAL choose times joint k1 = 5/32

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 · implies

Before

source roles = [5, 1, 1/2, 1/2]

After

derived = [5, 1/2, 1/16, 5/32]

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, S018

S020 · 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, S019

S021 · 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, S020

S022 · 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, S021

S023 · calculation

Before

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

After

CLT binomial k2 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, S022

S024 · calculation

Before

CLT binomial k2 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k2 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, S023

S025 · calculation

Before

CLT binomial ORIGINAL power product k2 := mul(1/4, 1/8)

After

CLT binomial ORIGINAL power product k2 = 1/32

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, S024

S026 · calculation

Before

CLT binomial ORIGINAL choose times joint k2 := mul(10, 1/32)

After

CLT binomial ORIGINAL choose times joint k2 = 5/16

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 · implies

Before

source roles = [5, 2, 1/2, 1/2]

After

derived = [10, 1/4, 1/8, 5/16]

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, S026

S028 · 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, S027

S029 · 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, S028

S030 · 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, S029

S031 · calculation

Before

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

After

CLT binomial k3 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, S030

S032 · calculation

Before

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

After

CLT binomial k3 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, S031

S033 · calculation

Before

CLT binomial ORIGINAL power product k3 := mul(1/8, 1/4)

After

CLT binomial ORIGINAL power product k3 = 1/32

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 · calculation

Before

CLT binomial ORIGINAL choose times joint k3 := mul(10, 1/32)

After

CLT binomial ORIGINAL choose times joint k3 = 5/16

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, S033

S035 · implies

Before

source roles = [5, 3, 1/2, 1/2]

After

derived = [10, 1/8, 1/4, 5/16]

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, S034

S036 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k4 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, S035

S037 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k4 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, S036

S038 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k4 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, S037

S039 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k4 success ORIGINAL power multiply4 = 1/16

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, S038

S040 · calculation

Before

CLT binomial k4 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k4 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, S039

S041 · calculation

Before

CLT binomial ORIGINAL power product k4 := mul(1/16, 1/2)

After

CLT binomial ORIGINAL power product k4 = 1/32

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, S040

S042 · calculation

Before

CLT binomial ORIGINAL choose times joint k4 := mul(5, 1/32)

After

CLT binomial ORIGINAL choose times joint k4 = 5/32

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, S041

S043 · implies

Before

source roles = [5, 4, 1/2, 1/2]

After

derived = [5, 1/16, 1/2, 5/32]

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, S042

S044 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k5 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, S043

S045 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k5 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, S044

S046 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k5 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, S045

S047 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply4 = 1/16

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, S046

S048 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply5 := mul(1/16, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply5 = 1/32

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, S047

S049 · implies

Before

source roles = [CLT binomial k5 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, S048

S050 · calculation

Before

CLT binomial ORIGINAL power product k5 := mul(1/32, 1)

After

CLT binomial ORIGINAL power product k5 = 1/32

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, S049

S051 · calculation

Before

CLT binomial ORIGINAL choose times joint k5 := mul(1, 1/32)

After

CLT binomial ORIGINAL choose times joint k5 = 1/32

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, S050

S052 · implies

Before

source roles = [5, 5, 1/2, 1/2]

After

derived = [1, 1/32, 1, 1/32]

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, S051

S053 · calculation

Before

CLT binomial ORIGINAL total mass sum0 := sum(0, 1/32)

After

CLT binomial ORIGINAL total mass sum0 = 1/32

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, S052

S054 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum0 := sum(0, 1/32)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum0 = 1/32

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, S053

S055 · calculation

Before

CLT binomial ORIGINAL total mass sum1 := sum(1/32, 5/32)

After

CLT binomial ORIGINAL total mass sum1 = 3/16

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, S054

S056 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum1 := sum(1/32, 5/32)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum1 = 3/16

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, S055

S057 · calculation

Before

CLT binomial ORIGINAL total mass sum2 := sum(3/16, 5/16)

After

CLT binomial ORIGINAL total mass sum2 = 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, S056

S058 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum2 := sum(3/16, 5/16)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum2 = 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, S057

S059 · calculation

Before

CLT binomial ORIGINAL total mass sum3 := sum(1/2, 5/16)

After

CLT binomial ORIGINAL total mass sum3 = 13/16

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, S058

S060 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum3 := sum(1/2, 5/16)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum3 = 13/16

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, S059

S061 · calculation

Before

CLT binomial ORIGINAL total mass sum4 := sum(13/16, 5/32)

After

CLT binomial ORIGINAL total mass sum4 = 31/32

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, S060

S062 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum4 := sum(13/16, 5/32)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum4 = 31/32

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, S061

S063 · calculation

Before

CLT binomial ORIGINAL total mass sum5 := sum(31/32, 1/32)

After

CLT binomial ORIGINAL total mass sum5 = 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, S062

S064 · implies

Before

source roles = [5, 1/2, 4, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]]]

After

derived = [1, 31/32, z_threshold=(4-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, S063

Reverse checks

  • S064 · 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.
  • S063 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum5 with exact rational arithmetic, not rounded decimals.
  • S062 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum4 with exact rational arithmetic, not rounded decimals.
  • S061 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum4 with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum3 with exact rational arithmetic, not rounded decimals.
  • S059 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum3 with exact rational arithmetic, not rounded decimals.
  • S058 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum2 with exact rational arithmetic, not rounded decimals.
  • S057 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum2 with exact rational arithmetic, not rounded decimals.
  • S056 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum1 with exact rational arithmetic, not rounded decimals.
  • S055 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum1 with exact rational arithmetic, not rounded decimals.
  • S054 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum0 with exact rational arithmetic, not rounded decimals.
  • S053 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum0 with exact rational arithmetic, not rounded decimals.
  • S052 · 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.
  • S051 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k5 with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k5 with exact rational arithmetic, not rounded decimals.
  • S049 · 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.
  • S048 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply5 with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S043 · 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.
  • S042 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k4 with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k4 with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute CLT binomial k4 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S035 · 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.
  • S034 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k3 with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k3 with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute CLT binomial k3 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute CLT binomial k3 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S027 · 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.
  • S026 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k2 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k2 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S019 · 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.
  • S018 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k1 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute CLT binomial k1 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
  • S011 · 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.
  • S010 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply5 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply4 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

(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, Yes, z_threshold=(4-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: ca1722c357d0957c195493ecce1f12188c391b069c126d5456c0a4c52952bd17
  • candidate record SHA-256: 80ebc48fa04969c70db212e76e8758915e85502edfb57ad5e89f8dc2e58bf33f
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON