GRE Probability and Counting
Probability questions look scary but lean on just a few rules. The GRE keeps the arithmetic simple; the whole game is setting the problem up right. Here are the rules that matter, the traps to avoid, and two tools to build the intuition.
The one formula
Every probability starts here: the probability of an event is the number of favorable outcomes divided by the total number of equally likely outcomes. It is always between 0 and 1. Drag the tool below to see probability as a fraction of an area, the mental model that makes the rest click.
Both events at once
Two independent events are each 50% likely. How likely is it that BOTH happen?
And, or, and the complement
Three rules cover almost every GRE probability question:
- And (independent events): multiply. P(A and B) = P(A) × P(B).
- Or: add, then subtract the overlap. P(A or B) = P(A) + P(B) − P(A and B).
- The complement: P(not A) = 1 − P(A). This is the key to “at least one” problems: find P(none) and subtract from 1, instead of adding up every case.
The single most common GRE trap is grinding out an “at least one” problem the long way. Reach for 1 − P(none) first, almost every time.
Counting: does order matter?
When you need the number of outcomes, one question decides everything: does order matter? If yes, use permutations; if no, use combinations. Play with the tool to feel the difference between arranging and choosing.
Permutations vs. combinations
From 5 runners you take the top 3. How does ARRANGING 3 of 5 (order matters) compare with CHOOSING 3 of 5 (order doesn't)?
Frequently asked questions
What probability topics are on the GRE?
Basic probability (favorable over total), the addition and multiplication rules, the complement (especially 'at least one') , and counting with permutations and combinations. The GRE keeps the arithmetic light; the difficulty is in setting the problem up correctly.
When do I add probabilities and when do I multiply?
Multiply for 'and' (both events happening, when independent): P(A and B) = P(A) × P(B). Add for 'or' (either event): P(A or B) = P(A) + P(B) − P(A and B), where the last term is 0 if the events cannot both happen.
What is the 'at least one' trick?
For 'at least one' problems, it is almost always faster to find the probability of none and subtract from 1: P(at least one) = 1 − P(none). Computing 'at least one' directly usually means adding many cases.
Permutations or combinations, how do I tell?
Ask whether order matters. If rearranging the same items counts as a new outcome (a race finish, a password), use permutations. If order does not matter (a committee, a handful of items), use combinations.
Drill probability with real GRE questions
Grezi gives you adaptive probability practice and an AI tutor that shows you where your setup went wrong, not just the answer.
More GRE Quant: all topics, formula sheet.