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Probability — ISEE Upper practice questions

On the ISEE Upper Quantitative Reasoning section, Probability covers chance, counting, and equally likely outcomes in multi-step word problems. A student has to model pairings, compare dice-sum likelihoods, and find the chance of drawing one type of item after using algebra to recover hidden counts. Questions often ask which statement is true about two events or ask for a probability after a short setup. Answer choices are typically fractions, comparative statements, or a whole number. Common traps include treating every dice sum as equally likely, mixing pair counts with permutations, and dividing before the parts of the collection are known.

  • Section: Quantitative Reasoning 37 questions on the paper
  • 89 practice questions in the app

Sample questions

Question 1

In a club, each pair of members shakes hands exactly once. If 10 new members joined the club and then each pair of members shook hands exactly once, the total number of handshakes would increase by 155. How many members were originally in the club?

  1. A9
  2. B10
  3. C11
  4. D15
Show answer

Answer: C — 11

If there were originally nn members, the 10 new members create 10n10n handshakes with original members and 10×92=45\dfrac{10\times9}{2}=45 handshakes among themselves. Thus 10n+45=15510n+45=155, so n=11n=11.

Question 2

Ava and Ben each roll two fair dice numbered 1 through 6 and record the sum. Ava scores a point if her sum is 7. Ben scores a point if his sum is 2 or 12. Which statement is true?

  1. AAva has a greater probability of scoring a point.
  2. BBen has a greater probability of scoring a point.
  3. CAva and Ben have the same probability of scoring a point.
  4. DThere is not enough information to determine who has the greater probability.
Show answer

Answer: A — Ava has a greater probability of scoring a point.

P(sum 7)=6/366/36; P(sum 2 or 12)=2/362/36. Ava is more likely.

Question 3

A music folder contains 45 sheets. The number of practice sheets is 5 more than the number of theory sheets, and the number of blank sheets is 3 times the number of theory sheets. One sheet is chosen at random. What is the probability that the chosen sheet is blank?

  1. A845\dfrac{8}{45}
  2. B815\dfrac{8}{15}
  3. C1345\dfrac{13}{45}
  4. D715\dfrac{7}{15}
Show answer

Answer: B — 815\dfrac{8}{15}

If there are tt theory sheets, then t+(t+5)+3t=45t+(t+5)+3t=45, so t=8t=8 and there are 24 blank sheets. Thus, the probability is 2445=815\dfrac{24}{45}=\dfrac{8}{15}.

Practice 89 Probability questions in the app

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