ACT · Math · Preparing for Higher Math: Statistics and Probability

Probability

Probability is part of Preparing for Higher Math: Statistics and Probability in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include probability from a count or table, two draws without replacement, independent events or draws with replacement and at least one (complement).

Common traps: using the wrong total in the denominator; using part-to-part odds as the probability; forgetting to simplify to an offered fraction.

Sample questions

Question 1
A fair coin is tossed, and a spinner with 5 equal sections numbered 1 to 5 is spun. What is the probability of getting heads and an even number?
- A.
  1/5
- B.
  1/4
- C.
  2/5
- D.
  9/10
Answer: A. 1/5
The events are independent, so multiply: P(heads) × P(even) = (1/2)(2/5) = 1/5. Choice B assumes even and odd are equally likely on the spinner; C ignores the coin; D adds the probabilities instead of multiplying.
Question 2
A class has 12 girls and 18 boys. Then 6 more girls join the class. If a student is now chosen at random, what is the probability that the student is a girl?
- A.
  1/3
- B.
  2/5
- C.
  1/2
- D.
  3/5
Answer: C. 1/2
After the change there are 18 girls among 36 students: 18/36 = 1/2. Choice A raises the total but not the girl count (12/36); B is the probability before the girls joined; D raises the girl count but keeps the old total (18/30).
Question 3
The table summarizes survey responses.
Improved Did not improve Total
Adult 43 26 69
Teen 50 60 110
Total 93 86 179
If a person is selected at random from those in the "Improved" category, what is the probability that the person is in the adult group?
- A.
  43/179
- B.
  69/179
- C.
  43/93
- D.
  43/69
Answer: C. 43/93
Restrict to the "Improved" column (total 93). Of these, 43 are in the adult group: 43/93.
Question 4
A field test for a plant disease was tried on 200 plants. Of the infected plants, 27 tested positive and 3 tested negative. Of the healthy plants, 18 tested positive and 152 tested negative. If a plant that tested positive is chosen at random, what is the probability that it is infected?
- A.
  27/200
- B.
  2/5
- C.
  3/5
- D.
  9/10
Answer: C. 3/5
Restrict to positive results: 27 + 18 = 45 plants, 27 of them infected, so 27/45 = 3/5. Choice A divides by all 200 plants; B is the healthy share of the positives; D swaps the condition, 27 of the 30 infected plants.
Question 5
In a town, 45% of households subscribe to a streaming service, 35% subscribe to a newspaper, and 15% subscribe to both. What is the probability that a randomly chosen household subscribes to at least one of the two?
- A.
  0.1575
- B.
  0.5
- C.
  0.65
- D.
  0.8
Answer: C. 0.65
P(A or B) = P(A) + P(B) − P(A and B) = 0.45 + 0.35 − 0.15 = 0.65. Choice A multiplies 0.45 × 0.35 as if the question asked for both and the events were independent; B subtracts the overlap twice; D does not subtract it.

The table summarizes survey responses.
	Improved	Did not improve	Total
Adult	43	26	69
Teen	50	60	110
Total	93	86	179

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