Conditional probability with tables and expected frequencies · Higher
| English | 中文 | Pinyin |
|---|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | 条件概率 | tiáo jiàn gài lǜ |
The probability a randomly selected student swims changes when we learn that the student cycles. The given condition changes which students are possible.
- The probability a randomly selected student swims changes when we learn that the student cycles. The given condition changes which students are possible.
- This lesson studies conditional probability 条件概率: A probability within the group selected by given information.
Choose the mathematical structure
- Conditioning restricts the denominator to the given group. In a table, divide the intersection count by the condition’s row/column total. P(A given B)=P(A and B)/P(B) when P(B)>0. Reversing the condition usually changes the denominator. Expected-frequency trees help interpret percentages without treating the two conditions as interchangeable.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines conditional probability?
A probability within the group selected by given information.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Using a class of 30 with 18 cyclists, 12 swimmers and 7 doing both, P(swim given cycle)=7/18, while P(cycle given swim)=7/12. The unconditional swim probability is 12/30=0.4. For an expected cohort of 1000, suppose 20% have a condition; a test is positive for 90% with it and 10% without it. Expected positive counts are 180 from 200 with the condition and 80 from 800 without it. Of 260 positive tests, the conditional proportion with the condition is 180/260=9/13≈0.6923, not 90%. These are stated illustrative model rates, not claims about a real diagnostic test.
Conditional probability with tables and expected frequencies
Conditioning restricts the denominator to the given group
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
From cyclists 18 and cyclist swimmers 7, find P(swim given cycle).
Restrict to 18 cyclists; 7 swim.
Test a tempting shortcut
- Given positive and positive given condition are different questions. Use the conditioned group total, not the grand total. A rare starting category can make false-positive counts significant even when the detection rate is high.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
P(A given B) always equals P(B given A). This claim is false. Explain which definition or assumption it violates.
Find total expected positives in the illustrative 1000-person model.
180 true positives plus 80 false positives.
P(A given B) always equals P(B given A).
Given positive and positive given condition are different questions. Use the conditioned group total, not the grand total. A rare starting category can make false-positive counts significant even when the detection rate is high.
Interpret a new situation
- AQA P9 Higher requires conditional calculations and interpretation using two-way tables, trees and Venn diagrams. Name the restricted group and retain expected counts until the final ratio.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the numerator of 180/260 in lowest terms.
Divide numerator and denominator by 20 to get 9/13.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.5. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A probability within the group selected by given information. Choose the relationship, show the method, check its assumptions and interpret the result.