PSAT 8/9 Math: Probability and Conditional Probability
Probability measures how likely an event is to happen, and the PSAT 8/9 tests this through basic calculations, real-world scenarios, and two-way tables. This free, complete lesson covers basic probability, probability as a decimal or percent, simplifying probability ratios, and two-way tables and conditional probability. 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 quiz below.
1. Basic Probability
Probability is calculated as: favorable outcomes ÷ total possible outcomes. It's often expressed as a fraction, but can also be written as a decimal or percent.
Worked Example: A club has 60 members, and 15 of them play piano. If a member is chosen at random, what's the probability they play piano?
15/60 = 1/4
1. A club has 60 members, and 15 of them play piano. If a member is chosen at random, what's the probability they play piano? (see worked example above)
Explanation: Probability = favorable / total = 15/60, which simplifies to 1/4.
2. A box has 30 balls: 12 red, 6 blue, and 12 green. What's the probability of drawing a blue ball?
Explanation: Probability = 6/30, which simplifies to 1/5.
3. A set of numbers is −5, 2, 7, 10. What's the probability that a randomly chosen value from this set is negative?
Explanation: Only 1 of the 4 values (−5) is negative, so the probability is 1/4.
2. Probability as a Decimal or Percent
A probability can be converted between fraction, decimal, and percent forms. Divide the fraction to get a decimal, then multiply by 100 to get a percent.
Worked Example: Out of 500 laptops, 10 are damaged. Find the probability that a randomly selected laptop is damaged, as a decimal.
10/500 = 0.02
1. Out of 500 laptops, 10 are damaged. Find the probability that a randomly selected laptop is damaged, as a decimal. (see worked example above)
Explanation: 10/500 = 0.02.
2. A factory finds that 18 out of every 90 items are defective. Express this probability as a percent.
Explanation: 18/90 = 0.2, which is 20%.
3. A survey of 150 customers finds that 90 are at least 30 years old. Express this probability as a percent.
Explanation: 90/150 = 0.6, which is 60%.
3. Simplifying Probability Ratios
A probability written as a fraction should usually be simplified to lowest terms by dividing the numerator and denominator by their greatest common factor.
Worked Example: A fruit basket has 10 fruits, 6 of which are apples. Find the probability of randomly selecting an apple, in simplest form.
6/10 = 3/5
1. A fruit basket has 10 fruits, 6 of which are apples. Find the probability of randomly selecting an apple, in simplest form. (see worked example above)
Explanation: The GCF of 6 and 10 is 2. Dividing: 6/2 = 3, 10/2 = 5, giving 3/5.
2. A street has 12 houses, and 3 of them have solar panels. Find the probability that a randomly selected house has solar panels, in simplest form.
Explanation: The GCF of 3 and 12 is 3. Dividing: 3/3 = 1, 12/3 = 4, giving 1/4.
3. A deck of 40 cards has 16 that are face cards. Find the probability of drawing a face card, in simplest form.
Explanation: The GCF of 16 and 40 is 8. Dividing: 16/8 = 2, 40/8 = 5, giving 2/5.
4. Two-Way Tables and Conditional Probability
A two-way table organizes data by two categories at once (such as gender and tablet ownership). To find a missing cell, use the fact that rows and columns must add up to their totals. Conditional probability asks for a probability within just one row or column of the table (a restricted group), not the whole table.
Worked Example: A table of 800 students shows 350 males total, with 220 males who own a tablet. How many males do not own a tablet?
350 − 220 = 130 males who do not own a tablet
1. A table of 800 students shows 350 males total, with 220 males who own a tablet. How many males do not own a tablet? (see worked example above)
Explanation: Total males minus males who own a tablet: 350 − 220 = 130.
2. A table shows 100 students total: 60 play sports, and of those, 24 are in the school band. What's the conditional probability that a student is in the band, given that they play sports?
Explanation: "Given that they play sports" restricts the group to just the 60 sports players, not all 100 students. The probability is 24/60, which simplifies to 2/5.
3. A table of 500 total people shows 280 who prefer tea, and among those, 112 also prefer reading over watching TV. Find the conditional probability that someone prefers reading, given that they prefer tea.
Explanation: "Given that they prefer tea" restricts the group to the 280 tea drinkers. The probability is 112/280, which simplifies to 2/5.
Common Mistakes to Avoid
- Dividing by the wrong total, always use the total number of possible outcomes in the denominator, not just a related but different number.
- Forgetting to simplify a probability fraction when the question or answer choices expect a reduced form.
- Using the full total instead of the restricted group in a conditional probability problem, "given that..." always signals a smaller denominator based on just that subgroup.
- Misreading a two-way table's rows and columns, double check which direction (row total or column total) a given number represents before using it.
- Forgetting to convert between fraction, decimal, and percent forms correctly when a question asks for a specific format.
Frequently Asked Questions
What's the difference between regular probability and conditional probability?
Regular probability divides by the total number of outcomes in the entire sample. Conditional probability divides by the total in just a smaller, specified subgroup (signaled by phrases like "given that" or "if we know").
How do I fill in a missing value in a two-way table?
Use the fact that each row and column must add up to its stated total. If a row's total and some of its individual values are known, subtract to find the missing one.
Where can I practice more problems like these?
The Probability and Conditional Probability quizzes in the PSAT 8/9 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 8/9. You can also browse the full PSAT 8/9 Math Question Bank.