A box contains 4 red balls, 5 blue balls, and 6 green balls. If two balls are drawn without replacement, what is the probability both are green?

The probability of drawing two green balls without replacement equals 1/5 when the first-draw probability (6 green out of 15 total = 6/15) is multiplied by the second-draw probability given the first was green (5 remaining green out of 14 total = 5/14), yielding the product (6/15) × (5/14) = 30/210 = 1/7 after simplification wait, this calculation yields 1/7, not 1/5. Let me recalculate carefully: Total balls = 4 + 5 + 6 = 15. P(first green) = 6/15 = 2/5. P(second green | first green) = 5/14. Combined probability = (6/15) × (5/14) = 30/210 = 1/7. However, option A is 1/7 and option B is 1/5. Based on standard probability calculation, the correct answer should be 1/7. But let me verify using combinations: Number of ways to choose 2 green from 6 = C(6,2) = 15. Total ways to choose any 2 from 15 = C(15,2) = 105. Probability = 15/105 = 1/7. Therefore, Correct Answer: A. 1/7

A. 1/7

This value correctly represents the probability through multiple verification pathways demonstrating compound probability mastery. Sequential probability method: P(first green) = 6/15 = 2/5; P(second green | first green) = 5/14; combined P = (6/15) × (5/14) = 30/210 = 1/7 after dividing numerator and denominator by 30. Combinatorial method: favorable outcomes = C(6,2) = 6!/(2!4!) = 15; total outcomes = C(15,2) = 15!/(2!13!) = 105; probability = 15/105 = 1/7 after dividing by 15. Decimal verification: 1 ÷ 7 ≈ 0.142857; sequential calculation 0.4 × 0.35714 ≈ 0.142857, confirming equivalence. Percentage interpretation: approximately 14.29% chance of drawing two green balls consecutively without replacement. This solution demonstrates comprehensive understanding of dependent probability including conditional probability application for sequential draws without replacement, combinatorial counting as alternative methodology, fraction simplification to lowest terms, and cross-verification through decimal approximation integrated competencies essential for statistical analysis, quality control sampling, genetics (allele inheritance probabilities), and game theory where accurate compound probability determination informs risk assessment and strategic decision-making under uncertainty with sequential dependencies.

B. 1/5

This probability equals 0.20 or 20%, exceeding the correct value by approximately 5.71 percentage points. The error likely originates from calculating with replacement instead of without replacement: P(both green with replacement) = (6/15) × (6/15) = 36/225 = 4/25 = 0.16 (not 0.20), or more plausibly from executing (6/15) × (6/15) = 36/225 then simplifying incorrectly to 1/5 (36/225 ÷ 9 = 4/25, not 1/5). Another plausible pathway involves calculating P(first green) = 6/15 = 2/5 = 0.4 then stopping and reporting this single-draw probability as the answer for two draws a fundamental misinterpretation of the problem requirements. Students might execute combinatorial method correctly (C(6,2)=15, C(15,2)=105) but then simplify 15/105 as 15÷15=1, 105÷15=7 yielding 1/7 correctly but mis-transcribe as 1/5 through digit substitution under time pressure. The 0.0571 excess probability represents meaningful overestimation that would substantially distort risk assessments in applied contexts like medical testing (false positive accumulation) or quality control (defect clustering) highlighting how replacement versus non-replacement distinctions fundamentally alter probability calculations with practical consequences for sampling methodology design and interpretation.

C. 2/7

This fraction equals approximately 0.2857 or 28.57%, nearly double the correct probability. The error likely originates from doubling the correct answer (1/7 × 2 = 2/7) through unstructured adjustment after partial calculation, or from executing sequential probability as (6/15) + (5/14) = 0.4 + 0.357 ≈ 0.757 then mis-simplifying to 2/7 through fraction approximation. Another plausible pathway involves combinatorial calculation where students compute favorable outcomes as 6 × 5 = 30 (ordered pairs) but total outcomes as C(15,2) = 105 (unordered), yielding 30/105 = 2/7 revealing inconsistent counting methodology mixing ordered and unordered approaches within the same calculation. This "mixed counting" error represents a critical conceptual deficiency where learners fail to maintain consistent outcome space definition (either both ordered or both unordered) throughout probability calculation a pervasive issue in combinatorial probability requiring explicit emphasis on outcome space coherence. Students might recognize 6 choices for first green and 5 for second but forget that total outcomes must also be counted as ordered pairs (15 × 14 = 210) to maintain consistency, yielding 30/210 = 1/7 when properly executed. The doubled probability magnitude reveals systematic counting inconsistency rather than arithmetic error, requiring explicit protocol reinforcement: "Define outcome space type first (ordered/unordered), then maintain consistency for both favorable and total counts."

D. 3/7

This value equals approximately 0.4286 or 42.86%, triple the correct probability and representing an extreme overestimation. The error likely originates from adding probabilities incorrectly: P(first green) + P(second green) = 6/15 + 5/14 ≈ 0.4 + 0.357 = 0.757 then approximating to 3/7 ≈ 0.4286 through unstructured reduction. Another plausible pathway involves calculating P(at least one green) instead of P(both green): 1 - P(no green) = 1 - [C(9,2)/C(15,2)] = 1 - 36/105 = 69/105 = 23/35 ≈ 0.657, not 3/7. Students might execute sequential probability as (6/15) × (6/14) = 36/210 = 6/35 ≈ 0.171 then mis-simplify to 3/7 through factor confusion (dividing numerator and denominator by 12 instead of 6). The value 3/7 also equals the proportion of green balls initially (6/15 = 2/5 = 0.4, not 3/7 ≈ 0.4286) suggesting possible confusion between single-draw probability and compound probability requirements. This distractor effectively identifies learners with severe conceptual gaps in compound probability who treat multi-stage events as single-stage or apply invalid operation combinations (addition instead of multiplication for independent/dependent sequential events) a foundational deficiency requiring explicit reinforcement of the multiplication principle for sequential probability with concrete tree diagram visualizations showing branch multiplication to build intuitive understanding of why probabilities compound multiplicatively rather than additively for consecutive successful outcomes.

Conclusion

The probability of 1/7 emerges through rigorous application of dependent probability principles for draws without replacement, verified through both sequential conditional probability multiplication and combinatorial counting with consistent outcome space definition (unordered pairs), simplified to lowest terms through greatest common divisor reduction. This problem reinforces critical probability competencies essential across quantitative domains: distinguishing between sampling with versus without replacement and their respective probability calculations, maintaining outcome space consistency (ordered versus unordered) throughout combinatorial counting, applying the multiplication principle for sequential dependent events, and simplifying complex fractions through systematic reduction protocols. Mastery of these integrated skills proves indispensable for statistics (sampling distribution theory), epidemiology (disease transmission modeling), quality assurance (defect probability in manufacturing batches), and finance (default correlation in loan portfolios) where accurate compound probability determination underpins risk quantification and decision-making under uncertainty with sequential dependencies. The distractors strategically target pervasive misconceptions including replacement/non-replacement confusion, inconsistent counting methodologies, and invalid probability operations (addition instead of multiplication) highlighting the necessity of explicit outcome space definition before calculation and mandatory verification through alternative methods (sequential versus combinatorial) to catch counting inconsistencies that fundamentally distort probability assessments with potentially severe consequences in risk-sensitive applications.