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Probability and counting — SHSAT practice questions

Probability and counting on the SHSAT Mathematics exam covers finding chances for one or more events and enumerating favorable outcomes, including a spinner paired with a die, marbles left after some are removed, and numbered tiles that meet one condition or another. A student has to write a ratio of desired outcomes to possible outcomes, multiply for independent and events, add for or events while avoiding double-counted overlap, and recompute the total after items are set aside. Items are multiple choice, with answers usually written as simplified fractions, and common traps reuse the original bag total or treat overlapping cases as disjoint.

  • Section: Mathematics 57 questions on the paper
  • 241 practice questions in the app

Sample questions

Question 1

A box contains 20 tiles numbered 1 through 20. If one tile is chosen at random, what is the probability that the number on the tile is divisible by 2 or divisible by 5?

  1. A12\dfrac12
  2. B35\dfrac35
  3. C23\dfrac23
  4. D34\dfrac34
Show answer

Answer: B — 35\dfrac35

There are 10 multiples of 2 and 4 multiples of 5 from 1 through 20. The 2 multiples of 10 were counted twice, so there are 10+4−2=1210+4-2=12 favorable tiles. The probability is 1220=35\dfrac{12}{20}=\dfrac35.

Question 2

A spinner is divided into 5 equal-sized sections: 3 blue, 1 red, and 1 green. Elena spins the spinner once and then separately rolls a standard six-sided die once. What is the probability that the spinner lands on blue AND the die shows a number greater than 4?

  1. A110\dfrac{1}{10}
  2. B215\dfrac{2}{15}
  3. C15\dfrac{1}{5}
  4. D1415\dfrac{14}{15}
Show answer

Answer: C — 15\dfrac{1}{5}

P(blue) = 3/5 and P(die greater than 4) = P(5 or 6) = 2/6 = 1/3. Since the spin and the roll are independent, multiply the probabilities: 3/5 x 1/3 = 1/5.

Question 3

A bag contains 45 marbles, 9 of which are red. Without looking, Malik removes 15 marbles from the bag, 4 of which turn out to be red, and sets them aside. The remaining marbles stay in the bag. What is the probability that a marble drawn at random from the marbles remaining in the bag will be red?

  1. A19\dfrac{1}{9}
  2. B215\dfrac{2}{15}
  3. C16\dfrac{1}{6}
  4. D310\dfrac{3}{10}
Show answer

Answer: C — 16\dfrac{1}{6}

After removing 15 marbles (4 of them red), 45 - 15 = 30 marbles remain in the bag, and 9 - 4 = 5 of those are red, so P(red) = 5/30 = 1/6.

Practice 241 Probability and counting questions in the app

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