Probability and Two-Way Tables · Sub-skill drill
Compound and Independent Events
When two events are independent, the probability that both occur is the product of their individual probabilities. When two events are mutually exclusive, the probability that either occurs is the sum of their individual probabilities. SAT questions occasionally combine these rules in two-step calculations, like the probability of drawing a red card and then a face card from a deck. Identify whether the events are independent and whether you want both or either before applying the rules.
How this sub-skill is tested on the SAT
When two events are independent, the probability that both occur is the product of their individual probabilities. When two events are mutually exclusive, the probability that either occurs is the sum of their individual probabilities. SAT questions occasionally combine these rules in two-step calculations, like the probability of drawing a red card and then a face card from a deck. Identify whether the events are independent and whether you want both or either before applying the rules.
This sub-skill sits inside the broader Probability and Two-Way Tables topic, which is part of the College Board's Problem Solving & Data Analysis content domain. Problem Solving and Data Analysis is where the SAT pretends to be the real world. Every question in this domain is wrapped in context: a recipe, a survey, a clinical trial, a lab measurement, a marketing report. The math itself is rarely harder than middle-school arithmetic — ratios, proportions, percentages, unit conversions, means, medians, scatter plots, two-way tables, and basic probability. What trips students up is the reading. The College Board has spent two decades calibrating these prompts to reward students who slow down on the setup and punish students who rush to compute. ScoreRe
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
-
A school surveyed students about a proposed schedule change. 21 boys and 42 girls said yes; 28 boys and 61 girls said no. Given that a randomly chosen student said yes, what is the probability the student is a boy?
- A 21/152
- B 63/152
- C 1/3
- D 2/3
Worked solution
Answer: C — 1/3
Conditional probability uses only the subset that said yes as the denominator. That subset is 21 + 42 = 63. Boys in that subset = 21. Probability = 21/63 = 1/3.
-
A school surveyed students about a proposed schedule change. 72 boys and 23 girls said yes; 70 boys and 80 girls said no. Given that a randomly chosen student said yes, what is the probability the student is a boy?
- A 72/245
- B 72/95
- C 23/95
- D 19/49
Worked solution
Answer: B — 72/95
Conditional probability uses only the subset that said yes as the denominator. That subset is 72 + 23 = 95. Boys in that subset = 72. Probability = 72/95 = 72/95.
-
A school surveyed students about a proposed schedule change. 53 boys and 79 girls said yes; 20 boys and 20 girls said no. Given that a randomly chosen student said yes, what is the probability the student is a boy?
- A 33/43
- B 53/172
- C 53/132
- D 79/132
Worked solution
Answer: C — 53/132
Conditional probability uses only the subset that said yes as the denominator. That subset is 53 + 79 = 132. Boys in that subset = 53. Probability = 53/132 = 53/132.
-
A school surveyed students about a proposed schedule change. 80 boys and 38 girls said yes; 37 boys and 50 girls said no. Given that a randomly chosen student said yes, what is the probability the student is a boy?
- A 40/59
- B 118/205
- C 16/41
- D 19/59
Worked solution
Answer: A — 40/59
Conditional probability uses only the subset that said yes as the denominator. That subset is 80 + 38 = 118. Boys in that subset = 80. Probability = 80/118 = 40/59.
-
A school surveyed students about a proposed schedule change. 36 boys and 80 girls said yes; 68 boys and 35 girls said no. Given that a randomly chosen student said yes, what is the probability the student is a boy?
- A 20/29
- B 12/73
- C 116/219
- D 9/29
Worked solution
Answer: D — 9/29
Conditional probability uses only the subset that said yes as the denominator. That subset is 36 + 80 = 116. Boys in that subset = 36. Probability = 36/116 = 9/29.
-
A school surveyed students about a proposed schedule change. 30 boys and 47 girls said yes; 66 boys and 45 girls said no. Given that a randomly chosen student said yes, what is the probability the student is a boy?
- A 47/77
- B 15/94
- C 30/77
- D 77/188
Worked solution
Answer: C — 30/77
Conditional probability uses only the subset that said yes as the denominator. That subset is 30 + 47 = 77. Boys in that subset = 30. Probability = 30/77 = 30/77.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.