SAT Probability and Conditional Probability: 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
Probability is the ratio of favorable outcomes to total possible outcomes: P(event) = favorable outcomes / total outcomes. A probability lies from 0 to 1, and P(not A) = 1 - P(A). Questions may use a two-way frequency table or a count scenario.
Conditional probability restricts the sample space to cases where a condition is already true: P(A | B) = P(A and B) / P(B). In a table, the denominator is the row or column total for B, not the grand total. Ask what outcomes remain after applying the condition before writing the fraction.
For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B). If events overlap, subtract P(A and B) once to avoid double-counting.
Expected value is the probability-weighted average of possible outcomes: multiply each outcome by its probability and add the products.
Key Rules & Formulas
Memorize these rules — they come up directly in practice 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?
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Correct answer: C. 3/10
Explanation
Total marbles = 5 + 3 + 2 = 10. P(green) = 3/10.
A standard six-sided die is rolled. What is the probability of rolling a number greater than 4?
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Correct answer: D. 1/3
Explanation
Numbers greater than 4 on a die: {5, 6}, which is 2 outcomes. P = 2/6 = 1/3.
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?
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Correct answer: A. 3/8
Explanation
P(Grade 9) = 30/80 = 3/8.
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?
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Correct answer: B. 11/20
Explanation
Condition: the student prefers Sports (total = 40). Of those, 22 are in Grade 10. P(Grade 10 | Sports) = 22/40 = 11/20.
A fair coin is flipped three times. What is the probability of getting exactly two heads?
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Correct answer: C. 3/8
Explanation
Total outcomes = 2³ = 8. Outcomes with exactly 2 heads: HHT, HTH, THH = 3 outcomes. P = 3/8.
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?
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Correct answer: D. 5/8
Explanation
Even numbers: {2, 4, 6, 8} (4 outcomes). Greater than 6: {7, 8} (2 outcomes). Overlap (even AND > 6): {8} (1 outcome). P(even OR > 6) = 4/8 + 2/8 − 1/8 = 5/8.
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?
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Correct answer: A. 9/20
Explanation
P(accepted AND interviewed) = 54/120 = 9/20.
A jar contains 4 red and 6 white marbles. Two marbles are drawn without replacement. What is the probability that both marbles are red?
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Correct answer: B. 2/15
Explanation
P(both red) = P(1st red) × P(2nd red | 1st red) = (4/10) × (3/9) = 12/90 = 2/15.
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?
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Correct answer: C. 1/5
Explanation
P(coffee or tea) = (120 + 80 − 40)/200 = 160/200 = 4/5. P(neither) = 1 − 4/5 = 1/5.
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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Correct answer: A. 3/4
Explanation
Students who play at least one sport = 18 + 12 − 6 = 24. Of these, 18 play soccer. P(soccer | at least one sport) = 18/24 = 3/4.
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Common Mistakes to Avoid
Use this checklist to catch common traps while working through Probability and Conditional Probability questions, then add your own patterns as you review missed answers.
- !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)
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, compare direct casework with the complement 1 − P(none); the complement often needs fewer calculations
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
Continue with this skill in the question bank, or test how it holds up under section timing before returning to the surrounding Problem Solving skills.