Make the possible outcomes clear. Try six probability questions, then work through the reasoning behind each answer.
Try the 6 questionsFree to practice. Answers and explanations included.
Before calculating, describe the outcome the question asks for. “Either,” “both,” “none,” and “at least one” each define a different event.
When outcomes are equally likely, probability is the number of favorable outcomes divided by the total. For “A or B,” an outcome in both groups still counts only once:
Without replacement, both the remaining total and the number of favorable items can change. For the playlist in Question 3, the second selection is from 14 remaining tracks, not the original 15.
Independent events satisfy
This set includes five single-answer questions and one quantitative comparison. Select one answer for each question, then open its explanation.
Question format: ETS Quantitative Reasoning overview.
A fair six-sided die is rolled once. What is the probability that the number rolled is even or greater than 4?
Select one answer.
Answer: A
The possible outcomes are 1, 2, 3, 4, 5, and 6. The even outcomes are 2, 4, and 6, and the outcomes greater than 4 are 5 and 6. Since 6 belongs to both sets, it is counted only once. Thus, the favorable outcomes are 2, 4, 5, and 6, so the probability is , or . The correct answer is Choice (A).
A shelf contains 32 books, of which 14 are fiction and 12 are paperbacks. If 5 of the books are both fiction and paperbacks, what is the probability that a book selected at random from the shelf is fiction or a paperback?
Select one answer.
Answer: B
Of the 32 books, 14 are fiction and 12 are paperbacks. The 5 books that are both fiction and paperbacks are included in both of these counts. Therefore, the number of books that are fiction or paperbacks is
A playlist consists of 12 instrumental tracks and 3 vocal tracks. The tracks are arranged in a random order, with each ordering equally likely. What is the probability that the first two tracks are both instrumental?
Select one answer.
Answer: E
The probability that the first track is instrumental is . If the first track is instrumental, 11 of the remaining 14 tracks are instrumental. Therefore, the probability that the first two tracks are both instrumental is
A container holds 16 tiles, of which 6 are triangles and 10 are circles. Three tiles are selected at random, one at a time, without replacement. What is the probability that none of the three selected tiles is a triangle?
Select one answer.
Answer: B
For none of the selected tiles to be triangles, all three selected tiles must be circles. The probability that the first tile is a circle is . After a circle is selected, 9 of the 15 remaining tiles are circles, and after a second circle is selected, 8 of the 14 remaining tiles are circles. Therefore, the required probability is
Events and are independent. If the probability that does not occur is and
Select one answer.
Answer: C
Since
An archive contains 160 grant applications. If one application is selected at random, the probability that it satisfies at least one of requirements and is , and the probability that it satisfies both requirements is . Quantity A: The number of applications that satisfy exactly one of requirements and Quantity B: Three times the number of applications that satisfy neither requirement nor requirement
Select one answer.
Answer: B
Of the 160 applications, the number that satisfy at least one requirement is
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