Evaluating a Compound Probability Benchmark Item
A district benchmark item for an eighth-grade probability unit reads: "A bag contains 3 red and 5 blue marbles. A marble is drawn, its color is recorded, and it is put back. Then a second marble is drawn. What is the probability that at least one marble is red?" The choices are 3/8, 9/64, 30/64, and 39/64. Across 120 students, 24% chose 39/64, 38% chose 30/64, 19% chose 9/64, 12% chose 3/8, and 7% chose other answers or left it blank. The item-writing team is reviewing the results. Several teachers report that English learners asked whether "at least one" allowed both marbles to be red. The team must decide whether the item is valid for the intended standard, which asks students to use the multiplication rule for independent events and to find probabilities of compound events. The team meets for 45 minutes and wants a decision it can defend to the principal.
1. Analyze the item's distractor pattern. What does each distractor reveal, and does the pattern support the claim that the item is measuring the intended construct?
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The correct value is 39/64: P(at least one red) = 1 − P(no red) = 1 − (5/8)(5/8) = 1 − 25/64 = 39/64, or equivalently P(exactly one red) + P(both red) = 30/64 + 9/64.
- 30/64 is P(exactly one red): 2(3/8)(5/8) = 30/64. The 38% choosing it are likely treating "at least one" as "exactly one," or they are counting the two ordered cases but omitting the both-red case. This is a meaningful compound-event error, not a random guess.
- 9/64 is P(both red): (3/8)(3/8). The 19% choosing it are likely interpreting "at least one" as "both," or they are stopping after one favorable case.
- 3/8 is the one-draw probability of red. The 12% choosing it are ignoring the second draw or misunderstanding the two-stage structure.
The pattern is not random: three distractors correspond to recognizable partial conceptions of compound probability, and the most common wrong answer is a coherent alternative interpretation of the language. That supports the claim that the item is measuring compound-event reasoning, but it also shows that the phrase "at least one" is doing extra linguistic work. The item is mathematically valid, yet its validity for the intended construct is weakened because a student can know the multiplication rule and still miss the item by misreading "at least one." The team should treat the 30/64 cluster as evidence about interpretation as well as about probability reasoning.
2. Recommend whether to retain, revise, or replace the item for future use. Support your recommendation with specific changes and an explanation of how the changes affect validity.
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Recommendation: revise, do not retain as written and do not discard. The construct-using the multiplication rule for independent events and combining outcomes-is worth assessing, but the current wording and options confound that construct with the interpretation of "at least one."
Specific revisions:
- Replace "at least one marble is red" with the more explicit "one or both marbles are red." This preserves the same mathematical event while reducing linguistic ambiguity.
- Add a distractor for 25/64, the probability of no red, so that the complement strategy is also visible in the option set.
- Keep 30/64 (exactly one red) and 9/64 (both red) as diagnostic distractors, since they reveal meaningful partial reasoning.
- Add a second item that asks directly for P(exactly one red) and another that asks for P(neither is red), so the team can separate students who can compute each part from students who can combine them.
- In the item stem, state that the draws are independent or that the marble is replaced, and keep that phrase in a consistent position.
Effect on validity: The revision reduces construct-irrelevant language demand while keeping the same probability content. The added complement distractor improves diagnostic value. Reporting the item alongside part-whole items gives the team evidence about whether the 30/64 group has a language issue or a part-whole issue. If the team cannot revise before the next administration, it should retain the item for instruction but flag it as ambiguous in the item bank.
3. Describe how the team should use the results instructionally for the 38% who chose 30/64, including one concrete task.
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The 30/64 group should be treated as having a partial compound-event scheme: they can compute the probability of exactly one red but have not yet coordinated the two mutually exclusive ways to satisfy "at least one." Instruction should make the sample space and the union of events visible rather than simply reteaching the formula.
Concrete task: Give students an 8-by-8 grid representing the 64 equally likely ordered outcomes for two draws with replacement. Label the first draw's outcomes R1, R2, R3, B1, B2, B3, B4, B5 across the top and the second draw's outcomes down the side. Have students shade all cells with at least one red, then count them. They should see 3×8 + 5×3 = 24 + 15 = 39 shaded cells, or equivalently 64 − 25 = 39. Then have them shade exactly one red in a different color and compare 30 with 39. Ask: "Which cells did the 30/64 answer leave out, and why must they be included?"
Follow with a verbal distinction: "at least one" means one or both; "exactly one" means one but not both. Have students write both events in set notation and compute each from the grid. The instructional goal is not just the correct fraction but the habit of checking whether all mutually exclusive cases have been included.