Probability and Conditional Probability: SAT Practice Questions & Study Guide
Questions asking you to find simple, compound, and conditional probabilities, often from two-way frequency tables.
Understanding Probability and Conditional Probability on the SAT
Probability is the ratio of favorable outcomes to total possible outcomes, and on the SAT it almost always appears in the context of a two-way frequency table or a simple count scenario. The basic formula is P(event) = (number of favorable outcomes) / (total outcomes). Probabilities are always between 0 and 1, and P(not A) = 1 − P(A).
Conditional probability is probability given that a specific condition is already known to be true: P(A | B) = P(A and B) / P(B). In table problems, this means your denominator is only the row or column total for the given condition—not the grand total of the entire table. This is the most critical distinction in SAT probability, and getting it right separates a correct answer from a trap. Before you write any fraction, ask: 'What is my new total sample space given the condition?'
Compound probability covers two events: 'and' vs. 'or.' For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events (cannot both happen), P(A or B) = P(A) + P(B). For non-mutually-exclusive events, P(A or B) = P(A) + P(B) − P(A and B) to avoid double-counting the overlap. The SAT rarely requires you to identify whether events are independent; instead it gives you counts directly from a table.
Expected value questions ask for the average outcome of a random process: E(X) = sum of (each outcome × its probability). These appear occasionally and follow directly from the probability formula—multiply each possible outcome by its probability and add the products.
Key Rules & Formulas
Memorize these rules — they come up directly in SAT questions.
Basic probability: P(A) = favorable outcomes / total outcomes
Bag with 4 red, 6 blue marbles: P(red) = 4/10 = 2/5
Complement rule: P(not A) = 1 − P(A)
P(not red) = 1 − 2/5 = 3/5
Conditional probability: P(A | B) = P(A and B) / P(B); in a table, denominator = row or column total for condition B
Of 200 students, 80 are seniors; 50 of those seniors play sports. P(plays sports | senior) = 50/80 = 5/8
Independent events 'and': P(A and B) = P(A) × P(B)
P(heads on flip 1 AND heads on flip 2) = 0.5 × 0.5 = 0.25
Non-exclusive events 'or': P(A or B) = P(A) + P(B) − P(A and B)
P(red or even) in a deck: must subtract P(red and even) to avoid double-counting
Probability and Conditional Probability Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
A bag contains 5 red, 3 green, and 2 blue marbles. If one marble is drawn at random, what is the probability of drawing a green marble?
A standard six-sided die is rolled. What is the probability of rolling a number greater than 4?
The table below shows the results of a survey of 80 students about their preferred after-school activity: | Sports | Reading | Total -----------+--------+---------+------ Grade 9 | 18 | 12 | 30 Grade 10 | 22 | 28 | 50 Total | 40 | 40 | 80 If one student is chosen at random from the 80 surveyed, what is the probability that the student is in Grade 9?
Using the table from the previous problem: | Sports | Reading | Total -----------+--------+---------+------ Grade 9 | 18 | 12 | 30 Grade 10 | 22 | 28 | 50 Total | 40 | 40 | 80 If a student is chosen at random from those who prefer Sports, what is the probability that the student is in Grade 10?
A fair coin is flipped three times. What is the probability of getting exactly two heads?
A spinner has 8 equal sections numbered 1 through 8. What is the probability of spinning a number that is either even or greater than 6?
Of 120 applicants to a program, 72 were accepted. Of the accepted applicants, 54 attended an interview. What is the probability that a randomly selected applicant was both accepted and attended an interview?
A jar contains 4 red and 6 white marbles. Two marbles are drawn without replacement. What is the probability that both marbles are red?
A survey of 200 employees found that 120 drink coffee, 80 drink tea, and 40 drink both. What is the probability that a randomly selected employee drinks neither coffee nor tea?
In a class of 30 students, 18 play soccer and 12 play basketball. 6 students play both. A student is selected at random. Given that the student plays at least one sport, what is the probability that the student plays soccer?
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Common Mistakes to Avoid
These are the most frequent errors students make on Probability and Conditional Probability questions. Knowing them in advance prevents costly point losses.
- !Using the table's grand total as the denominator in a conditional probability problem instead of the appropriate row or column total
- !Confusing P(A | B) with P(B | A)—these are generally not equal
- !Adding probabilities for 'and' scenarios instead of multiplying (and vice versa for 'or')
- !Forgetting to subtract the overlap when computing P(A or B) for non-mutually-exclusive events
- !Treating 'at least one' problems by listing all cases rather than using the complement: P(at least one) = 1 − P(none)
SAT Strategy Tips: Probability and Conditional Probability
For any table-based probability problem, physically circle or underline the condition before you decide which total to put in the denominator
When a problem says 'given that' or 'if we know that,' that is always a conditional probability—make sure your denominator reflects the restricted sample space
For 'at least one' scenarios, compute 1 − P(none) instead of summing multiple cases; it is almost always faster
Check that your final probability is between 0 and 1—if it isn't, you've made an arithmetic error
Other Problem Solving & Data Analysis Subtopics
Ratios, Rates, and Proportional Relationships
Questions asking you to set up proportional equations, convert units, and scale quantities in real-world contexts.
Percentages and Percent Change
Questions covering percent conversions, percent increase and decrease, and multi-step percentage problems with real-world price or quantity contexts.
Statistics, Data Interpretation, and Distributions
Questions requiring you to calculate and interpret measures of center and spread, read graphs and tables, and understand the shape of data distributions.
Statistical Inference and Study Design
Questions testing whether you can identify what a study's design allows you to validly conclude, including generalizability and causality.
Master Probability and Conditional Probability on the SAT
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