Compute probability and conditional probability from a two-way table or area model.
Use the grand total for regular probability and the subgroup total for conditional probability.
核心知识
In a two-way table, a regular probability uses the grand total as the denominator. A conditional probability uses a row or column total as the denominator.
深入理解
Two-way tables organize data by two categories — say, grade level across the top and sport preference down the side. Every SAT probability question involving a two-way table asks you to pull the right cell value (numerator) and the right total (denominator).
For an unconditional probability like "What fraction of all students prefer basketball?", your denominator is the grand total in the bottom-right corner.
For a conditional probability like "Among juniors, what fraction prefers basketball?", your denominator is the juniors row total. You've restricted your universe to juniors only.
The key skill is matching the wording to the correct row or column total. Phrases like "given that," "among," "of those who," and "if we know" all signal a conditional probability — they tell you which subtotal to use as the denominator.
Area models work the same way. The full rectangle represents probability 1, and each sub-region represents a joint probability. Conditional probability means you zoom into one strip and ask what fraction of that strip is shaded.
分步讲解
- Read the question and identify what event you're looking for (numerator).
- Determine whether the question is conditional — look for 'given that,' 'among,' 'of those who.'
- If unconditional, the denominator is the grand total. If conditional, the denominator is the relevant row or column total.
- Find the cell value for the numerator and the correct total for the denominator. Divide and simplify.
常见误解
- Using the grand total as the denominator when the question specifies a subgroup (conditional probability).
- Swapping which variable is the condition and which is the event — 'probability of A given B' is different from 'probability of B given A.'
- Reading the wrong cell because the row and column labels were confused.
示例解析
A survey of 200 students recorded whether each student is in 9th or 10th grade and whether each prefers online or in-person classes. Of the 90 ninth-graders, 54 prefer online classes. Of the 110 tenth-graders, 44 prefer online classes. What is the probability that a randomly selected student who prefers online classes is a tenth-grader?
选择一个答案查看解析