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 bag contains 5 red, 7 blue, and 8 green marbles. One marble is selected at random. What is the probability that it is blue?
- A.
  1/3
- B.
  7/20
- C.
  8/20
- D.
  7/13
Answer: B. 7/20
Total = 5 + 7 + 8 = 20. P(blue) = 7/20. A (7/13) uses 7 out of non-blue total; C (1/3) estimates instead of computing exactly; D (8/20) uses the green count.
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 walking-tour guide had 80 visitors in one week. Of them, 46 joined the old-town tour, 38 joined the harbour tour and 15 joined both. A visitor is chosen at random from those who joined at least one tour. What is the probability that the visitor joined both tours?
- A.
  11/80
- B.
  5/28
- C.
  3/16
- D.
  5/23
Answer: D. 5/23
Visitors on at least one tour: 46 + 38 − 15 = 69. So P(both | at least one) = 15/69 = 5/23 (D). C divides by all 80 visitors, ignoring the condition; B uses 46 + 38 = 84, counting the overlap twice; A is the probability that a visitor joined neither tour, 11/80.
Question 4
On Rania's drive to school, the probability that the first traffic light is green when she reaches it is 0.6 and the probability that the second light is green is 0.5, independently. What is the probability that exactly one of the two lights is green?
- A.
  0.3
- B.
  0.5
- C.
  0.8
- D.
  1.1
Answer: B. 0.5
Exactly one: green then not green, 0.6 × 0.5 = 0.3, plus not green then green, 0.4 × 0.5 = 0.2, for 0.5 (B). A counts only the first case; C is 'at least one green', 1 − 0.4 × 0.5, which includes both green; D adds 0.6 and 0.5.
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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