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Data analysis and probability — ISEE Upper practice questions

Data analysis and probability on the ISEE Upper Mathematics Achievement section covers counting, set diagrams, chance, and expected value. Students must find the probability that at least one of two conditions holds on two rolls of a number cube, sort three-club overlap to count those in exactly one group, and compute an expected number of points from a discrete reward. Questions are multi-step multiple choice. Common traps include double-counting outcomes that meet both conditions, treating pairwise overlap as if nobody belongs to all three clubs, and forgetting that remaining participants take the leftover outcome. Answer choices usually look like nearby whole numbers or simple fractions produced by counting mistakes.

  • Section: Mathematics Achievement 47 questions on the paper
  • 321 practice questions in the app

Sample questions

Question 1

A fair six-sided number cube is rolled twice. What is the probability that the sum of the two rolls is 9 or the product of the two rolls is 20?

  1. A118\dfrac{1}{18}
  2. B112\dfrac{1}{12}
  3. C19\dfrac{1}{9}
  4. D16\dfrac{1}{6}
Show answer

Answer: C — 19\dfrac{1}{9}

The pairs with sum 9 are (3,6),(4,5),(5,4),(3,6),(4,5),(5,4), and (6,3)(6,3). The pairs with product 20 are (4,5)(4,5) and (5,4)(5,4), which are already included. There are 4 favorable outcomes out of 36, so the probability is 436=19\dfrac{4}{36}=\dfrac{1}{9}.

Question 2

A reward system gives exactly one of three point awards. The probability of receiving 2 points is 20%20\%, and the probability of receiving 10 points is 25%25\%. All remaining participants receive 6 points. What is the expected number of points received?

  1. A2.90
  2. B6.00
  3. C6.20
  4. D8.90
Show answer

Answer: C — 6.20

The probability of receiving 6 points is 100%−20%−25%=55%100\%-20\%-25\%=55\%. Thus, the expected value is 2(0.20)+6(0.55)+10(0.25)=6.202(0.20)+6(0.55)+10(0.25)=6.20.

Question 3

At a school, 27 students belong to the robotics club, 23 belong to the debate club, and 19 belong to the art club. The pairwise overlaps are 10, 8, and 7, and each includes the 3 students who belong to all three clubs. How many students belong to exactly one of the three clubs?

  1. A16
  2. B19
  3. C28
  4. D47
Show answer

Answer: C — 28

The pair-only regions contain (10−3)+(8−3)+(7−3)=16(10-3)+(8-3)+(7-3)=16 students. Since there are 69 club memberships total, the number in exactly one club is 69−2(16)−3(3)=2869-2(16)-3(3)=28.

Practice 321 Data analysis and probability questions in the app

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