A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability it is not blue?

The probability of drawing a non-blue marble equals 7/10 when the favorable outcomes (5 red + 2 green = 7 non-blue marbles) are divided by the total possible outcomes (5 + 3 + 2 = 10 marbles), yielding the fraction 7/10 that represents the likelihood of selecting any marble except blue from the bag.

A. 3/10

This fraction equals precisely the probability of drawing a blue marble (3 blue ÷ 10 total = 3/10) rather than its complement, revealing a fundamental confusion between an event and its complement. Students selecting this option likely calculated the blue probability correctly but failed to recognize the problem requested "not blue" rather than "blue" a critical reading comprehension error with profound implications in probability contexts where complement relationships (P(not A) = 1 - P(A)) form essential problem-solving tools. The error might also originate from miscounting non-blue marbles as 3 instead of 7 through arithmetic error (5 + 2 = 7 miscomputed as 3) or visual misreading of the bag composition. This distractor effectively identifies learners who execute computational steps correctly but misinterpret problem requirements a deficiency requiring explicit emphasis on careful question analysis before solution execution, particularly in probability where subtle wording changes (e.g., "is blue" versus "is not blue") completely reverse solution approaches and outcomes.

B. 7/10

This value correctly represents the non-blue probability through multiple verification pathways demonstrating conceptual and computational mastery. Direct counting method: non-blue marbles = 5 red + 2 green = 7; total marbles = 10; probability = 7/10. Complement method: P(blue) = 3/10; P(not blue) = 1 - 3/10 = 7/10. Decimal verification: 7 ÷ 10 = 0.7 or 70% likelihood. Percentage interpretation: 70% of marbles are non-blue, confirming intuitive expectation that majority selection yields non-blue outcome given blue constitutes minority (30%). Reduction check: 7 and 10 share no common factors beyond 1, confirming fraction is in simplest form. This solution demonstrates comprehensive understanding of probability fundamentals including outcome enumeration, sample space definition, complement relationships, fraction simplification, and multiple representation fluency (fraction, decimal, percentage) integrated competencies essential for statistical reasoning, risk assessment, game theory, and decision analysis where accurate probability determination informs strategic choices under uncertainty.

C. 1/2

This fraction equals 0.5 or 50%, suggesting students approximated the non-blue count as 5 instead of 7 (perhaps counting only red marbles while omitting green) or executed 10 ÷ 2 = 5 then assumed 5 favorable outcomes. Another plausible error pathway involves averaging probabilities (P(red) = 5/10, P(green) = 2/10; average = 3.5/10 ≈ 1/2 through unstructured approximation). The 0.2 deficit from correct probability (0.7 - 0.5 = 0.2) represents exactly two marbles' worth of probability mass corresponding precisely to the omitted green marbles. This distractor proves particularly instructive for diagnosing incomplete outcome enumeration, a pervasive error in probability where learners identify some but not all favorable cases, often due to categorical oversight (recognizing red as non-blue but forgetting green also qualifies). Real-world implications abound: medical testing (omitting disease subtypes in risk calculation), quality control (missing defect categories), or survey analysis (excluding demographic segments) where incomplete enumeration produces dangerously optimistic probability assessments with potentially severe consequences for decision-making accuracy.

D. 2/5

This fraction equals 0.4 or 40%, corresponding precisely to the green marble probability alone (2 green ÷ 10 total = 2/5) rather than the combined non-blue probability. The error likely originates from recognizing green as non-blue but completely omitting red marbles from favorable outcomes a categorical partiality error where students fixate on one non-blue category while ignoring others. Another plausible pathway involves misreading "not blue" as "green" through visual scanning error or cognitive fixation on the last-listed color. The 0.3 deficit from correct probability (0.7 - 0.4 = 0.3) equals exactly the red marble probability (5/10 = 0.5) minus an adjustment revealing systematic omission rather than random miscalculation. This distractor effectively targets learners with fragmented categorical reasoning who fail to recognize that "not blue" encompasses all non-blue categories collectively rather than a single alternative category a conceptual deficiency requiring explicit instruction on set complementarity and exhaustive enumeration principles fundamental to probability theory and logical reasoning across mathematical domains.

Conclusion

The 7/10 probability emerges through rigorous outcome enumeration recognizing that "not blue" includes all marbles except the three blue ones, validated through both direct counting and complement methods ensuring result integrity. This problem reinforces critical probabilistic literacy competencies essential across quantitative domains: distinguishing events from their complements, executing exhaustive outcome enumeration without categorical omission, applying the fundamental probability formula (favorable/total), and leveraging complement relationships (P(not A) = 1 - P(A)) as computational shortcuts. Mastery of these integrated skills proves indispensable for statistical analysis (p-value interpretation), risk management (failure probability assessment), gaming strategy (odds calculation), and scientific experimentation (control group analysis) where accurate probability determination prevents costly misjudgments in uncertainty quantification. The distractors strategically target pervasive error patterns including event-complement confusion, partial enumeration, and categorical fixation highlighting the necessity of systematic outcome listing protocols (e.g., explicitly writing "non-blue = red + green") before probability calculation to guard against oversight errors that compromise analytical validity in increasingly complex probabilistic scenarios.