Conditional probability with tables and expected frequencies · Higher
הוצג| English | משמעות |
|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | הסתברות תנאי |
The given condition changes which students are possible.
- The probability a randomly selected student swims changes when we learn that the student cycles.
- This lesson studies conditional probability 条件概率: A probability within the group selected by given information.
The method
- 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.
In the 1000-person model, why is the proportion of positive tests that are correct only 180/260?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Conditioning reverses the question: P(positive given condition)=90% answers a different question from P(condition given positive). The rare-condition group is small, so false positives from the large healthy group dilute the positive pool to 180/260≈69%.
A worked case
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). Give it to 3 significant figures.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Restrict to 18 cyclists; 7 swim.
Where it goes wrong
- 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.
Not true: P(A given B) equals P(B given A). The two conditional probabilities have different denominators, so they are usually different.
Find total expected positives in the illustrative 1000-person model.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
180 true positives plus 80 false positives.
P(A given B) equals P(B given A).
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The two conditional probabilities have different denominators, so they are usually different.
Applying it
- 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.
Find the numerator of 180/260 in lowest terms.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Divide numerator and denominator by 20 to get 9/13.
Match each conditional quantity to its denominator.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
The condition selects the group whose total becomes the denominator; reversing the condition usually changes the value.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.5.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Conditional probability: A probability within the group selected by given information.
- Conditioning restricts the denominator to the given group.
- Watch for: given positive and positive given condition are different questions.