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
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 2
Twelve cards are numbered 1 through 12. Two cards are drawn at random without replacement. What is the probability that both numbers are multiples of 3?
- A.
  1/11
- B.
  1/9
- C.
  4/33
- D.
  20/33
Answer: A. 1/11
Four cards (3, 6, 9, 12) are multiples of 3: P = (4/12)(3/11) = 12/132 = 1/11. Choice B draws with replacement, (1/3)²; C lowers the total on the second draw but keeps 4 favorable cards; D adds the two probabilities instead of multiplying.
Question 3
Ten people ride an elevator in an office tower: 6 work in sales and 4 work in IT. Two of them are chosen at random, one after the other, for a short survey. What is the probability that both work in sales?
- A.
  3/10
- B.
  1/3
- C.
  9/25
- D.
  3/5
Answer: B. 1/3
After one salesperson is chosen, 5 of the remaining 9 are in sales, so P = (6/10)(5/9) = 1/3. Choice A keeps 10 in the second denominator; C treats the draws as independent, (6/10)²; D is the probability for the first person only.
Question 4
A sports club has 250 members, and 140 of them are adults. In all, 90 members prefer swimming to other sports, and 35 of those 90 are adults. A member who is not an adult is chosen at random. What is the probability that this member prefers swimming?
- A.
  11/50
- B.
  7/22
- C.
  1/2
- D.
  11/18
Answer: C. 1/2
Non-adults: 250 − 140 = 110. Non-adult swimmers: 90 − 35 = 55. So 55/110 = 1/2. Choice A divides by all 250 members; B uses the adult swimmers over the non-adults; D swaps the condition, 55 out of the 90 swimmers.
Question 5
Two fair six-sided dice are rolled. What is the probability that the sum is 8 or at least one die shows a 6?
- A.
  55/1296
- B.
  1/3
- C.
  7/18
- D.
  4/9
Answer: C. 7/18
Sum 8: 5 outcomes. At least one 6: 11 outcomes. Both: (2, 6) and (6, 2). So 5 + 11 − 2 = 14 of 36, or 7/18. Choice A multiplies 5/36 by 11/36 as if the events were independent; B subtracts the overlap twice (12/36); D does not subtract it (16/36).

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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