Probability Word Problem Terms Guide

A guide to translating common English phrases into mathematical equations for probability problems.

1. Inequality Keywords

These terms define ranges of possibilities rather than exact numbers.

Phrase	Symbol	Meaning	Example (Rolling a 6-sided die)
At least	≥	The value or anything higher. (Minimum is x)	"At least 4" You win if you roll 4, 5, or 6.
At most	≤	The value or anything lower. (Maximum is x)	"At most 2" You win if you roll 1 or 2.
More than	>	Strictly higher than the value.	"More than 3" You win if you roll 4, 5, or 6.
Less than	<	Strictly lower than the value.	"Less than 5" You win if you roll 1, 2, 3, or 4.
No fewer than	≥	Same as "At least".	"No fewer than 2" You win if you roll 2, 3, 4, 5, or 6.
No more than	≤	Same as "At most".	"No more than 3" You win if you roll 1, 2, or 3.

2. Exact & Inclusive/Exclusive Terms

Exactly (=): The specific number, no more, no less.
Example: "Exactly two heads" in 3 flips = {HHT, HTH, THH}.
Inclusive (Between): Includes the endpoints.
Example: "Between 2 and 5 inclusive" = {2, 3, 4, 5}.
Exclusive (Between): Does not include the endpoints.
Example: "Between 2 and 5" usually implies {3, 4}.

3. Logic Keywords (And / Or / Not)

"OR" (Union ∪)

Action: Addition (Subtract any overlap).
Formula: P(A or B) = P(A) + P(B) - P(A and B)
Example: "King OR Heart" (Add Kings + Hearts, subtract King of Hearts).

"AND" (Intersection ∩)

Action: Multiplication.
Formula: P(A and B) = P(A) × P(B) (if independent).
Example: "Roll 6 AND flip Heads" (1/6 × 1/2 = 1/12).

"NOT" (Complement A')

Action: Subtraction from 1.
Formula: P(Not A) = 1 - P(A)
Example: Useful for "At least one" problems (1 - Probability of None).

4. Selection Keywords

Order Matters (Permutation): "Arranged", "Line up", "Code", "Schedule".
Order Does Not Matter (Combination): "Selected", "Group", "Committee", "Hand".
With Replacement: Item is put back; probabilities stay the same.
Without Replacement: Item is kept; total items decrease by 1.

Summary Cheat Sheet

Term	Symbol/Math	Quick Translation
At least	≥	Minimum limit
At most	≤	Maximum limit
Or	+	Add
And	×	Multiply
Not	1 - x	Subtract from total