SAT · Math · Problem-Solving and Data Analysis

Probability and conditional probability

Probability and conditional probability is part of Problem-Solving and Data Analysis in the SAT 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 conditional probability from a table, probability of a random selection, complement and union and independent events.

Common traps: using the grand total as the denominator; using the wrong row or column total; reversing the condition.

Sample questions

Question 1
A box contains 40 pens: 14 blue, 10 red, and the rest black. If one pen is selected at random, what is the probability that it is black?
Enter your own answer.
Question 2
The table summarizes survey responses.
Bus Car Total
Morning-shift 9 24 33
Evening-shift 58 25 83
Total 67 49 116
If a person is selected at random from all respondents, what is the probability that the person is in the evening-shift group and in the "Bus" category?
- A.
  1/2
- B.
  58/83
- C.
  83/116
- D.
  58/67
Answer: A. 1/2
There are 58 such people out of 116 total: 58/116 = 1/2.
Question 3
A bag contains 5 red, 3 blue, and 4 green marbles. Two marbles are drawn at random, one after the other, without replacement. What is the probability that both marbles are green?
- A.
  1/11
- B.
  1/9
- C.
  4/33
- D.
  1/3
Answer: A. 1/11
The first draw is green with probability 4/12. With one green gone, 3 greens remain among 11 marbles, so the second is green with probability 3/11. The product is (4/12)(3/11) = 12/132 = 1/11. B uses (4/12)² as if the first marble were replaced; C reduces the total to 11 but keeps 4 greens; D is the probability of a single green draw.
Question 4
In a random sample of 200 of a city's 50,000 households, 34 households own an electric car. Based on the sample, about how many households in the city do not own an electric car?
- A.
  166
- B.
  8,500
- C.
  41,500
- D.
  49,966
Answer: C. 41,500
In the sample 200 − 34 = 166 households, or 83%, do not own one, and 0.83 × 50,000 = 41,500 (C). A is the sample count, not scaled up; B estimates the households that do own one; D subtracts only the 34 sampled owners from 50,000.
Question 5
A point is chosen at random inside a square with sides of 10 units. What is the probability that the point is within 5 units of the centre of the square?
- A.
  π/16
- B.
  1 − π/4
- C.
  1/2
- D.
  π/4
Answer: D. π/4
The points within 5 units of the centre form a circle of radius 5, which fits inside the square: area 25π out of 100, so the probability is π/4 (D). A uses 5 as the diameter; B is the probability of being farther than 5 units; C compares lengths, 5/10, instead of areas.

The table summarizes survey responses.
	Bus	Car	Total
Morning-shift	9	24	33
Evening-shift	58	25	83
Total	67	49	116

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