A bag contains 8 red marbles and 12 blue marbles. What is the probability of drawing a red marble?

The probability of drawing a red marble equals 2/5 when the favorable outcomes (8 red marbles) are divided by the total possible outcomes (8 + 12 = 20 marbles), yielding the simplified fraction 8/20 = 2/5 that represents the likelihood of selecting a red marble from the bag in a single random draw.

A. 2/5

This fraction correctly represents the probability through systematic outcome enumeration and simplification verified through multiple approaches. Direct calculation: P(red) = 8/20 = 2/5 after dividing numerator and denominator by their greatest common divisor 4. Decimal verification: 2 ÷ 5 = 0.4; 8 ÷ 20 = 0.4, confirming equivalence. Percentage interpretation: 40% of marbles are red, aligning with intuitive expectation given red constitutes 8 of 20 total items. Complement verification: P(blue) = 12/20 = 3/5; P(red) + P(blue) = 2/5 + 3/5 = 1, confirming exhaustive and mutually exclusive outcomes. Reduction check: 2 and 5 share no common factors beyond 1, confirming simplest form. This solution demonstrates comprehensive understanding of basic probability including outcome enumeration, sample space definition, fraction simplification, complement relationships, and multiple representation fluency integrated competencies essential for statistical reasoning, risk assessment, game theory, and decision analysis where accurate single-event probability determination forms the foundation for increasingly complex probabilistic modeling requiring precise likelihood quantification under conditions of uncertainty.

B. 3/5

This fraction equals precisely the probability of drawing a blue marble (12/20 = 3/5) rather than red, revealing a fundamental event misidentification where students calculated the complement probability but failed to recognize the problem requested red rather than blue. Students might have correctly computed 12/20 = 3/5 but then misread the question's color specification during answer selection a critical reading comprehension error with profound implications in probability contexts where subtle wording changes completely reverse solution requirements. Another plausible pathway involves calculating the ratio of blue to red (12:8 = 3:2) then misinterpreting this ratio as the blue probability 3/5 without recognizing the denominator must represent total outcomes (20), not just the comparison group (8). The symmetric relationship between options A and B (2/5 versus 3/5 summing to 1) reveals this distractor targets the most common error in two-category probability problems: confusing an event with its complement a deficiency requiring explicit emphasis on careful question analysis before computation, particularly in probability where event specification determines numerator selection while total count determines denominator, with complement relationships serving as valuable verification tools rather than solution substitutes.

C. 1/2

This value equals 0.5 or 50%, suggesting students approximated the red count as 10 instead of 8 (perhaps rounding to nearest multiple of 5) or executed 20 ÷ 2 = 10 then assumed 10 favorable outcomes. Another plausible error pathway involves averaging the counts (8 + 12 = 20 ÷ 2 = 10) then assuming equal probability despite unequal quantities a fundamental misconception that probability depends on category count equality rather than proportional representation. Students might recognize two color categories and incorrectly assume binary outcomes imply 50-50 probability without considering quantity disparities, revealing categorical reasoning deficiency where learners treat categories as equally likely regardless of population distribution. The 0.1 deficit from correct probability (0.5 - 0.4 = 0.1) represents exactly two marbles' worth of probability mass corresponding precisely to the difference between assumed 10 red marbles and actual 8. This distractor proves particularly instructive for diagnosing incomplete outcome enumeration influenced by categorical symmetry assumptions rather than quantitative reality a pervasive error with real-world implications in medical testing (assuming disease prevalence equals non-prevalence), quality control (assuming defect rates are 50%), or survey analysis (assuming demographic parity) where inaccurate base rate assumptions produce dangerously optimistic probability assessments compromising decision accuracy in risk-sensitive contexts.

D. 2/3

This fraction equals approximately 0.666 or 66.7%, corresponding precisely to the ratio of blue to red marbles (12:8 = 3:2 inverted to 2:3 then misread as 2/3) rather than red to total. The error likely originates from calculating the red-to-blue ratio (8:12 = 2:3) then misinterpreting this ratio's second term as the probability denominator rather than recognizing probability requires favorable over total (8/20), not favorable over unfavorable (8/12 = 2/3). Students might execute 8 ÷ 12 = 2/3 correctly for the odds ratio but then conflate odds with probability a critical conceptual distinction where odds represent favorable-to-unfavorable ratio while probability represents favorable-to-total ratio, with conversion requiring denominator adjustment (odds a:b → probability a/(a+b)). The 0.267 excess above correct probability (0.667 - 0.400 = 0.267) reveals systematic ratio misinterpretation rather than random miscalculation diagnostically valuable for identifying learners who lack differentiation between related probabilistic measures (probability versus odds versus ratio), a deficiency requiring explicit contrast between these concepts with concrete examples demonstrating how 2:3 odds converts to 2/5 probability through denominator expansion (2/(2+3) = 2/5), building robust conceptual frameworks preventing measure conflation that distorts risk perception in gambling, insurance, and medical decision-making contexts.

Conclusion

The probability 2/5 emerges through rigorous outcome enumeration recognizing that probability requires favorable outcomes divided by total outcomes not favorable over unfavorable or category count assumptions with simplification to lowest terms validated through complement verification ensuring exhaustive outcome coverage. This problem reinforces critical probabilistic literacy competencies essential across quantitative domains: distinguishing between probability (favorable/total), odds (favorable/unfavorable), and simple ratios; executing exhaustive outcome enumeration without categorical symmetry assumptions; applying fraction simplification protocols; and leveraging complement relationships (P(A) + P(not A) = 1) as verification tools rather than solution substitutes. Mastery of these integrated skills proves indispensable for statistical analysis (base rate understanding), medical diagnostics (prevalence-aware testing interpretation), finance (risk probability assessment), and artificial intelligence (Bayesian reasoning) where accurate single-event probability determination prevents base rate neglect and supports sound decision-making under uncertainty. The distractors strategically target pervasive error patterns including event-complement confusion, categorical symmetry assumptions overriding quantitative reality, and odds-probability conflation highlighting the necessity of explicit outcome space definition protocols ("What constitutes success? What constitutes all possibilities?") before probability calculation to guard against conceptual errors that compromise analytical validity in increasingly complex probabilistic scenarios requiring precise likelihood quantification.