Original question and givens
S={0,…,2}, F=P(S), masses=[0, 1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[], B=[1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — S={0,…,2}, F=P(S), masses=[0, 1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[], B=[1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.
Step-by-step derivation
S001 · implies
Before
source roles = [qf-density10-conditional-masses-pair, density10_conditional_masses_pair, finite_normalized_probability]
After
derived = [[P_intersection, P_B]]
Conditions: none
Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources.
Step identifiers and dependencies
Rule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001
S002 · calculation
Before
Conditional ORIGINAL ALL numerator weights sum := sum(0, 1, 2)
After
Conditional ORIGINAL ALL numerator weights sum = 3
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
source roles = [[0, 1, 2], 3]
After
derived = [3, [[0, 0/1], [1, 1/3], [2, 2/3]], 3]
Conditions: none
On S=Fin(n) with its full powerset,each singleton has mass m_i/D,all m_i>=0 and total D>0.These FOUR families REQUIRE an actual normalized law;reject an invalid law rather than return No and repair its weights.
Step identifiers and dependencies
Rule: Conditional_FULL_normalized_finite_law_not_candidate_repair · Uses: G001, S002
S004 · implies
Before
source roles = [B, [1], 3]
After
derived = [[1]]
Conditions: none
FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound.
Step identifiers and dependencies
Rule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S003
S005 · calculation
Before
Conditional B ORIGINAL included index1 running0 := sum(0, 1)
After
Conditional B ORIGINAL included index1 running0 = 1
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [Conditional B, [1], [0, 1, 2]]
After
derived = [1]
Conditions: none
For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment.
Step identifiers and dependencies
Rule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S005
S007 · calculation
Before
Conditional ORIGINAL B mass over D := div(1, 3)
After
Conditional ORIGINAL B mass over D = 1/3
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
source roles = [[1], [0, 1, 2], 3]
After
derived = [1, 1/3, Yes]
Conditions: none
P(B)>0 means b>0,not merely B nonempty.A nonempty event containing only null labels cannot be conditioned on.Divide by b ONLY under the given positive-mass premise.
Step identifiers and dependencies
Rule: Conditional_STRICT_positive_B_mass_not_merely_nonempty · Uses: G001, S007
S009 · implies
Before
source roles = [A, [], 3]
After
derived = [[]]
Conditions: none
FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound.
Step identifiers and dependencies
Rule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S008
S010 · implies
Before
source roles = [Conditional intersection, [], [0, 1, 2]]
After
derived = [0]
Conditions: none
For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment.
Step identifiers and dependencies
Rule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S009
S011 · calculation
Before
Conditional ORIGINAL intersection mass over D := div(0, 3)
After
Conditional ORIGINAL intersection mass over D = 0
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [[], [1], [], [0, 1, 2], 3]
After
derived = [0/1, 1/3]
Conditions: none
These TWO fields are P(A intersection B) and P(B),each divided by D,not the conditional ratio i/b.Keep their declared order.
Step identifiers and dependencies
Rule: Conditional_MASSES_pair_separate_unconditional_denominator_D · Uses: G001, S011