QUANTITATIVE REASONING · GMAT-Q-PROB

Probability Foundations

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical relationship

P (E) = \frac{favorable outcomes}{all equally likely outcomes}

Translate the condition

List the equally likely outcomes or use the stated counts to define the sample space. For sequential draws, update the counts when there is no replacement.

Test the result

Check that the probability is between zero and one and that the denominator matches the available outcomes at that step.

Think Like a GMAT Test-Taker

Define the sample space and whether draws are with replacement before multiplying probabilities.

Faster Methods

Use ordered draws when the sequence of outcomes matters.

For “exactly one,” add the disjoint red-blue and blue-red cases.

Worked example 1

A fair selection is made from 10 equally likely items, of which 2 are marked. What is the probability of a marked item, as a percentage?

P = \frac{2}{10} = \frac{1}{5} = 20%

There are 2 marked items among 10 equally likely items, so 2 divided by 10 is 0.2, or 20 percent.

Worked example 2

A bag holds 5 red and 4 blue tokens. Two are selected without replacement. What is the probability of at least one red token?

P (at least one red) = 1 - \frac{4}{9} \cdot \frac{3}{8} = \frac{5}{6}

The complement is two blue tokens. It is easier to compute this single excluded event than to add all orders containing red. Update the denominator after the first draw, then subtract from one.

Worked example 3

A fair coin is tossed three times. What is the probability of exactly one head?

P = \frac{( 1 3 )}{2 ^{3}} = \frac{3}{8}

There are three possible positions for the head. Each complete sequence has probability one eighth. Exactly one excludes the all-tail outcome and every sequence with two or three heads.

Common Traps

Do not keep the original denominator after a draw without replacement.
Do not omit a valid order when the event can occur in multiple orders.