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Prerequisite Diagnostic

Allow 20–30 minutes. These questions assess arithmetic, sets, and counting from earlier modules. Probability knowledge is not required. Explain each answer before opening the feedback.

  1. Simplify (3/20)/(3/20 + 1/10) and identify the denominator's value.
  2. List every ordered two-character string over {H,T}. Group them by number of H characters. How many strings enter each group?
  3. Ten machines include three marked machines. How many unordered groups of four are possible? How many contain exactly one marked machine?
  4. A has 8 elements, B has 7, and their overlap has 3. Find the size of their union and explain the subtraction.
  5. A batch has 100 tasks taking 10 ms each and 900 taking 100 ms each. Find the mean task time; explain why averaging 10 and 100 directly answers a different question.
  6. Optional coding bridge: write a loop that counts H characters in each string from question 2. If syntax is unfamiliar, use a hand table first and revisit the programming foundations before the executable labs.
Answers and targeted repair
  1. Denominator=1/4, ratio=3/5. If division of fractions is difficult, review ratios and units, then try (2/15)/(2/15+1/5) (answer 2/5).
  2. HH,HT,TH,TT; group counts for 0,1,2 H are 1,2,1. If order was lost, review Cartesian products, then list all length-three strings (8).
  3. C(10,4)=210; exactly one marked gives C(3,1)*C(7,3)=105. Review combinations versus permutations; transfer: choose two marked and two unmarked (63).
  4. 8+7-3=12; the overlap was counted twice. Review set union and intersection, then explain why disjoint sets need no subtraction.
  5. (100*10+900*100)/1000=91 ms. A 50/50 batch would instead average to 55 ms. Review weighted sums and what the denominator counts.
  6. Iterate over the four strings, sum one for each H, and compare with the hand table. The lab uses loops, functions, dictionaries, CSV, and the standard library; no statistics package is required.

Repair a missed prerequisite and solve its changed case before the matching cluster. This is a placement aid, not a probability mastery score. Record which items needed assistance; no arbitrary total score overrides a specific missing prerequisite.