Compute probabilities (including conditional probability) from tables/contexts.
In conditional probability, shrink the sample space first and then count the favorable outcomes.
核心知识
Probability is favorable outcomes divided by possible outcomes, but the denominator must match the condition in the question. Words such as "given," "among," and "of those" mean restrict the sample space first.
深入理解
Rule: In basic probability, the denominator is the total number of possible outcomes in the full sample space. In conditional probability, the denominator changes because you are no longer looking at the full sample space; you are looking only at the cases that meet the condition.
On ACT Math, the most common error is using the total from the whole table when the question asks about one row, one column, or one subgroup. Read the condition carefully, then rebuild the denominator from that restricted group before counting favorable outcomes.
分步讲解
- Identify the condition, if there is one.
- Restrict the sample space to only the cases that satisfy that condition.
- Count favorable outcomes within that restricted group.
- Write probability as favorable over possible and simplify if needed.
常见误解
- Using the grand total when the question asks for a conditional probability.
- Switching the direction of the condition, such as using
𝑃 ( 𝐴 ∣ 𝐵 ) 𝑃 ( 𝐵 ∣ 𝐴 ) - Counting the right favorable cases but from the wrong subgroup.
示例解析
A school recorded whether students attended tutoring and whether they passed a quiz.
| Passed | Did not pass | |
|---|---|---|
| Tutoring | 18 | 6 |
| No tutoring | 12 | 14 |
What is the probability that a randomly chosen student passed the quiz, given that the student attended tutoring?
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