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Probability and sample spaces — ACT practice questions

Probability and sample spaces on the ACT Mathematics test covers finding the chance of an event by comparing favorable outcomes with a clearly defined sample space. A student has to count equally likely outcomes, use combinations when a committee is chosen at random, treat geometric probability as a ratio of areas when a point is selected from a region, and find leftover cases such as tiles that are neither of two colors. Items are multiple choice with five answer choices, usually written as simplified fractions. Common traps include using permutations instead of combinations, confusing radius with area, and forgetting to exclude already counted categories.

  • Section: Mathematics 45 questions on the paper
  • 57 practice questions in the app

Sample questions

Question 1

A 3-member committee is chosen randomly from 4 seniors and 3 juniors. What is the probability that the committee contains exactly 2 seniors and 1 junior?

  1. A635\dfrac{6}{35}
  2. B37\dfrac{3}{7}
  3. C1835\dfrac{18}{35}
  4. D2435\dfrac{24}{35}
Show answer

Answer: C — 1835\dfrac{18}{35}

There are (73)=35\binom{7}{3}=35 possible committees and (42)(31)=18\binom{4}{2}\binom{3}{1}=18 favorable committees. The probability is 1835\dfrac{18}{35}.

Question 2

A circular flower bed with radius 6 feet lies entirely within a circular lawn with radius 10 feet. A point is selected at random from the lawn. What is the probability that the point is NOT in the flower bed?

  1. A925\dfrac{9}{25}
  2. B25\dfrac{2}{5}
  3. C1625\dfrac{16}{25}
  4. D45\dfrac{4}{5}
Show answer

Answer: C — 1625\dfrac{16}{25}

The bed covers π(6)2=36π\pi(6)^2=36\pi of the lawn's π(10)2=100π\pi(10)^2=100\pi square feet, so the point misses the bed with probability 100π−36π100π=1625\dfrac{100\pi-36\pi}{100\pi}=\dfrac{16}{25}.

Question 3

A collection contains 28 ceramic tiles. Of these tiles, 9 are blue and 6 are green; all the remaining tiles are white. One tile will be randomly selected. What is the probability that the selected tile will be neither blue nor green?

  1. A628\dfrac{6}{28}
  2. B928\dfrac{9}{28}
  3. C1328\dfrac{13}{28}
  4. D1528\dfrac{15}{28}
Show answer

Answer: C — 1328\dfrac{13}{28}

There are 9+6=159+6=15 blue or green tiles, so 28−15=1328-15=13 tiles are neither color. The probability is therefore 1328\dfrac{13}{28}.

Practice 57 Probability and sample spaces questions in the app

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