GMAT Quant
Probability
Use complements, multiplication, and inclusion-exclusion to model common probability questions.
Base definition
Basic probability is favorable outcomes divided by total outcomes.
This direct ratio works when outcomes are equally likely and the sample space is clear.
Complement
"At least one" is often fastest through the complement.
P(at least one) = 1 - P(none). Counting every acceptable case separately is usually slower.
Example: at least one head in 3 fair coin flips.
- P(no heads)
- (1/2)^3 = 1/8
- P(at least one head)
- 1 - 1/8
- Result
- 7/8
Independent AND
For independent events joined by AND, multiply.
If one event does not change the probability of the other, P(A and B) = P(A)*P(B).
Dependent events
Without replacement, later probabilities change.
After the first draw, both the favorable count and the total count may be different.
Example: 4 red and 6 blue; draw 2 red without replacement.
- First red
- 4/10
- Second red
- 3/9
- Both red
- (4/10)*(3/9) = 2/15
Inclusion-exclusion
For OR, subtract the overlap once.
P(A or B) = P(A) + P(B) - P(A and B). Otherwise outcomes in both events are counted twice.
Mini application
Try this: a fair die is rolled twice. What is P(at least one 5)?
Use the complement before scrolling.
Solution
The probability is 11/36.
P(no 5) = (5/6)^2 = 25/36, so 1 - 25/36 = 11/36.
Remember this
Match the language to the operation.
At least one -> complement. Independent AND -> multiply. OR -> add and subtract overlap.