Probability and Counting

Concise answer

Probability questions reduce to counting outcomes and combining events correctly: multiply along independent events, adjust for overlap when adding, use the complement for "at least one," and pick permutations only when order genuinely matters.

Definitions

Probability: A number between 0 and 1 measuring how likely an event is; for equally likely outcomes it is favorable outcomes divided by total outcomes.
Complement: The event that $A$ does not occur, written $A'$; its probability is $1 - P(A)$.
Independent events: Events where the occurrence of one does not change the probability of the other.
Dependent events: Events where the occurrence of one changes the probability of the other, as when drawing without replacement.
Permutation: An arrangement of items in which order matters.
Combination: A selection of items in which order does not matter.

Intuition

Multiplying probabilities mirrors counting in stages: if the first stage happens in some fraction of cases and the second in some fraction of those, the joint fraction is the product — which is exactly why dependence matters, since the second fraction can shift after the first outcome.

Every combination corresponds to $r!$ permutations of the same chosen items, so the combination count is the permutation count divided by $r!$ — order is 'counted out.'

Concept walkthrough

For a situation with equally likely outcomes, $P(A)$ is the number of outcomes in $A$ divided by the total number of outcomes, so probability questions are counting questions in disguise. Probabilities always lie between 0 and 1, and an event and its complement split the total: $P(A') = 1 - P(A)$ . The complement rule is the standard shortcut for "at least one" questions, because "at least one" has many cases but its complement — "none" — has exactly one.

Compound events combine by two rules. The multiplication rule gives $P(A \text{ and } B) = P(B) \cdot P(A|B)$ , which simplifies to $P(A) \cdot P(B)$ when the events are independent — successive coin flips, or draws with replacement. Drawing without replacement makes events dependent: the second probability must be updated to reflect the first draw. The addition rule gives $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ ; the subtraction removes the double-counted overlap and disappears only when the events are mutually exclusive.

Counting questions on the GRE come down to whether order matters. Arranging $r$ of $n$ distinct items in order is a permutation, counted by $\tfrac{n!}{(n-r)!}$ ; selecting $r$ of $n$ items when order is irrelevant is a combination, counted by $\tfrac{n!}{r!\,(n-r)!}$ . ETS lists elementary probability, conditional probability, and counting methods such as combinations and permutations among the data analysis topics of the Quantitative Reasoning measure.

After this page, you should be able to

Compute a basic probability as favorable outcomes over equally likely total outcomes.
Use the complement rule to handle "at least one" and "not" questions in one subtraction.
Decide whether events are independent or dependent, and multiply probabilities accordingly.
Choose between a combination and a permutation by asking whether order matters, and evaluate the factorial formulas.

Formulas and assumptions

Complement rule

P(A') = 1 - P(A)

Variables

P(A): the probability that event A occurs
P(A'): the probability that event A does not occur

Assumptions

A' contains exactly the outcomes not in A, so the two events cover all outcomes without overlap.

Multiplication rule for independent events

P(A \text{ and } B) = P(A) \cdot P(B)

Variables

P(A), P(B): the probabilities of the individual events
P(A and B): the probability that both occur

Assumptions

A and B are independent; for dependent events use $P(A \text{ and } B) = P(B) \cdot P(A|B)$ with the updated conditional probability.

Permutations and combinations of n items taken r at a time

{}_{n}P_{r} = \dfrac{n!}{(n-r)!} \qquad {}_{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}

Variables

n: the number of distinct available items
r: the number of items arranged (permutation) or selected (combination)

Assumptions

Items are distinct and used at most once.
Use the permutation count only when order matters; the combination count divides out the r! orderings of each selection.

Worked example

At least one head in three flips

A fair coin is flipped $3$ times. What is the probability of getting at least one head?

1The complement of "at least one head" is "no heads," meaning all three flips are tails.
2The flips are independent, so multiply: $P(\text{all tails}) = \tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{8}$ .
3Apply the complement rule: $P(\text{at least one head}) = 1 - \tfrac{1}{8} = \tfrac{7}{8}$ .
4Check by counting: $2^3 = 8$ equally likely outcomes, and only $TTT$ has no heads, so $7$ of $8$ outcomes qualify.

The probability is $\tfrac{7}{8}$ .

Common traps

Adding probabilities of events that can happen together without subtracting the overlap $P(A \text{ and } B)$ .
Multiplying probabilities of dependent events as if they were independent — after drawing without replacement, the denominator (and often the numerator) of the next probability changes.
Using a permutation count for a committee-style selection where order does not matter, which overcounts by a factor of $r!$ .
Confusing mutually exclusive with independent: mutually exclusive events with nonzero probabilities are never independent, because one occurring forces the other's probability to 0.

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Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE General Test Quantitative Reasoning Overview — ETS. Accessed 2026-08-02. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
Introductory Statistics 2e, Section 3.1: Terminology — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 3.2: Independent and Mutually Exclusive Events — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 3.3: Two Basic Rules of Probability — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Contemporary Mathematics, Section 7.2: Permutations — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Contemporary Mathematics, Section 7.3: Combinations — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.

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Last source check: 2026-08-02
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Recheck the ETS Quantitative Reasoning content page and the OpenStax statistics and counting references before each major GRE preparation cycle.

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Probability and Counting

Concise answer

Definitions

Intuition

Concept walkthrough

After this page, you should be able to

Formulas and assumptions

Complement rule

Multiplication rule for independent events

Permutations and combinations of n items taken r at a time

Worked example

At least one head in three flips

Common traps

Related pages and practice

Sources

Review and maintenance