PSAT/NMSQT Math: Probability
Probability questions on the PSAT/NMSQT ask how likely a specific outcome is, using everything from simple counts to tables and algebraic expressions. This free, complete lesson covers basic probability, complement probability and reading tables, probability with variables and finding quantities from a known probability, and the fundamental counting principle. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. Basic Probability
Probability is the number of favorable outcomes divided by the total number of possible outcomes: P = favorable/total. It's always a number between 0 (impossible) and 1 (certain), often expressed as a fraction, decimal, or percentage.
Worked Example: A choir has 60 members, 15 of whom are sopranos. If one member is randomly selected, find the probability they are a soprano.
P(soprano) = 15/60 = 1/4
1. A choir has 60 members, 15 of whom are sopranos. Find the probability a randomly selected member is a soprano. (see worked example above)
Explanation: P(soprano) = 15/60, which simplifies to 1/4.
2. A jar has 5 red marbles, 3 green marbles, and 2 blue marbles. Find the probability of drawing a green marble.
Explanation: Total marbles = 5 + 3 + 2 = 10. P(green) = 3/10.
3. A shipment of 2,000 headphones contains 30 defective units. Find the probability of selecting a defective headphone, as a decimal.
Explanation: P(defective) = 30/2000 = 0.015.
2. Complement Probability and Reading Tables
The complement of an event is everything that event does NOT include; its probability is 1 minus the probability of the original event: P(not A) = 1 − P(A). Probability questions using a two-way table require carefully identifying the correct row and column totals before dividing.
Worked Example: A bag has 20 marbles: 6 red, 5 yellow, and the rest a mix of other colors. Find the probability a randomly selected marble is neither red nor yellow.
Red or yellow: 6 + 5 = 11 marbles. Neither: 20 − 11 = 9 marbles.
P(neither red nor yellow) = 9/20
1. A bag has 20 marbles: 6 red, 5 yellow, and the rest other colors. Find the probability a randomly selected marble is neither red nor yellow. (see worked example above)
Explanation: Red or yellow = 11 marbles. The complement, neither red nor yellow, is 20 − 11 = 9 marbles, giving 9/20.
2. A gym has 80 members. A table shows 32 members are under 30 years old. Find the probability a randomly selected member is at least 30 years old.
Explanation: Members at least 30 = 80 − 32 = 48. P(at least 30) = 48/80 = 3/5.
3. A table of customer purchases shows 120 total customers, and 45 did not buy a drink. What is the probability a randomly selected customer did NOT buy a drink?
Explanation: The probability of not buying a drink is read directly: 45 out of 120 total, or 45/120 = 3/8.
3. Probability with Variables and Finding Quantities
Some probability problems use a variable to represent an unknown count instead of a specific number, the probability is still set up as favorable outcomes over total outcomes, just written algebraically. If you're given a probability and a total, you can solve for the unknown favorable count.
Worked Example: A shelf has m mystery books and 15 science books. Express the probability of randomly selecting a mystery book.
P(mystery) = m / (m + 15)
1. A shelf has m mystery books and 15 science books. Express the probability of randomly selecting a mystery book. (see worked example above)
Explanation: Favorable outcomes (mystery books) = m. Total books = m + 15. P(mystery) = m / (m + 15).
2. A bag has 60 total marbles. If the probability of drawing a red marble is 2/5, how many red marbles are in the bag?
Explanation: (2/5) × 60 = 24 red marbles.
3. Find the probability of rolling a 7 on a fair 18-sided die (faces numbered 1 to 18).
Explanation: There is exactly 1 favorable outcome (rolling a 7) out of 18 equally likely faces, giving 1/18.
4. The Fundamental Counting Principle
The fundamental counting principle states that if one event can happen in a ways and a second (independent) event can happen in b ways, then both events together can happen in a × b ways. This extends to any number of independent choices made in sequence.
Worked Example: How many unique 4-digit codes can be created using digits 0-9, if repeated digits are allowed?
Each of the 4 positions has 10 possible digits, independently:
10 × 10 × 10 × 10 = 10,000 codes
1. How many unique 4-digit codes can be created using digits 0-9, with repeated digits allowed? (see worked example above)
Explanation: 10 choices for each of 4 positions, independently: 10 × 10 × 10 × 10 = 10,000.
2. A restaurant offers 3 appetizers, 5 entrees, and 2 desserts. How many different 3-course meals (one from each category) are possible?
Explanation: Multiply the choices for each independent category: 3 × 5 × 2 = 30.
3. A license plate uses 2 letters followed by 3 digits (letters and digits can repeat). How many different license plates are possible?
Explanation: Each of the 5 positions (2 letters, 3 digits) is an independent choice: 26 × 26 for the letters, times 10 × 10 × 10 for the digits.
Common Mistakes to Avoid
- Dividing by the wrong total. The denominator in a probability must always be the total number of possible outcomes, not just one category or subgroup.
- Forgetting that complement probabilities must add to 1. P(A) + P(not A) always equals 1, a useful check on your work.
- Misreading rows versus columns in a two-way table. Always confirm which total (row, column, or grand total) the question is actually asking you to divide by.
- Adding instead of multiplying in a counting principle problem. Independent, sequential choices are always multiplied together, not added.
- Forgetting to account for repeated values allowed (or not allowed) when counting arrangements. Whether repetition is permitted changes whether each position has the same number of choices.
Frequently Asked Questions
Can a probability ever be greater than 1 or less than 0?
No. A valid probability always falls between 0 (impossible) and 1 (certain), inclusive. If your calculated answer falls outside this range, you've made an arithmetic or setup error.
When should I use the complement instead of calculating directly?
Use the complement whenever the event you actually want is harder to count directly than "everything else." Finding P(not A) and subtracting from 1 is often much faster than counting every case in A itself.
Where can I practice more problems like these?
The Probability quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.