6 GRE Probability Questions with Explanations

This set includes five single-answer questions and one quantitative comparison. Select one answer for each question, then open its explanation.

Question 1

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.

\frac{2}{3}

\frac{5}{6}

\frac{1}{2}

\frac{1}{3}

\frac{1}{6}

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Answer: A ( $\frac{2}{3}$ )

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 $\frac{4}{6}$ , or $\frac{2}{3}$ . The correct answer is Choice (A).

Question 2

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.

\frac{1}{2}

\frac{21}{32}

\frac{5}{32}

\frac{13}{16}

\frac{11}{32}

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Answer: B ( $\frac{21}{32}$ )

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 $14 + 12 - 5 = 21$ . Since each book is equally likely to be selected, the probability is $\frac{21}{32}$ . Thus, the correct answer is Choice (B).

Question 3

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.

\frac{11}{14}

\frac{4}{5}

\frac{34}{35}

\frac{16}{25}

\frac{22}{35}

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Answer: E ( $\frac{22}{35}$ )

The probability that the first track is instrumental is $\frac{12}{15}$ . 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 $(\frac{12}{15})$ ${}(\frac{11}{14})$ ${}= \frac{22}{35}$ . The correct answer is Choice E.

Question 4

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.

\frac{5}{8}

\frac{3}{14}

\frac{125}{512}

\frac{1}{28}

\frac{11}{14}

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Answer: B ( $\frac{3}{14}$ )

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 $\frac{10}{16}$ . 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 $\left(\frac{10}{16}\right)\left(\frac{9}{15}\right)\left(\frac{8}{14}\right)$ ${}= \frac{3}{14}$ . The correct answer is Choice B.

Question 5

Events $J$ and $K$ are independent. If the probability that $J$ does not occur is $\frac{3}{5}$ and $P(K) = \frac{1}{3}$ , what is $P(J \text{ or } K)$ ?

Select one answer.

\frac{11}{15}

\frac{2}{15}

\frac{3}{5}

\frac{1}{5}

\frac{7}{15}

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Answer: C ( $\frac{3}{5}$ )

Since $P(J \text{ does not occur})$ ${}= \frac{3}{5}$ , $P(J)$ ${}= 1 - \frac{3}{5}$ ${}= \frac{2}{5}$ . Because $J$ and $K$ are independent, $P(J \text{ and } K)$ ${}= (\frac{2}{5})$ ${}(\frac{1}{3})$ ${}= \frac{2}{15}$ . By the addition rule, $P(J \text{ or } K)$ ${}= P(J)$ ${}+ P(K)$ ${}- P(J \text{ and } K)$ ${}= \frac{2}{5}$ ${}+ \frac{1}{3}$ ${}- \frac{2}{15}$ ${}= \frac{6}{15}$ ${}+ \frac{5}{15}$ ${}- \frac{2}{15}$ ${}= \frac{9}{15}$ ${}= \frac{3}{5}$ .

Question 6

Quantitative comparison

An archive contains 160 grant applications. If one application is selected at random, the probability that it satisfies at least one of requirements $R$ and $S$ is $\frac{13}{16}$ , and the probability that it satisfies both requirements is $\frac{3}{10}$ . Quantity A: The number of applications that satisfy exactly one of requirements $R$ and $S$ Quantity B: Three times the number of applications that satisfy neither requirement $R$ nor requirement $S$

Select one answer.

AQuantity A is greater.BQuantity B is greater.CThe two quantities are equal.DThe relationship cannot be determined from the information given.

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Answer: B (Quantity B is greater.)

Of the 160 applications, the number that satisfy at least one requirement is $160 \times \left(\frac{13}{16}\right)$ ${}= 130$ , and the number that satisfy both requirements is $160 \times \left(\frac{3}{10}\right)$ ${}= 48$ . Therefore, $130 - 48 = 82$ applications satisfy exactly one requirement. Also, $160 - 130 = 30$ applications satisfy neither requirement, so Quantity B is $3 \times 30 = 90$ . Since $90$ is greater than $82$ , Quantity B is greater.

GRE probability questions

Start with the event you are counting.

Count each outcome once.

Check what changes after a selection.

Separate independence from overlap.