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With A the even numbers and B the numbers above 13 from 1 to 19, the intersection holds three values. What are P(A given B) and P(B given A)?
ABoth equal 3/19, so they are the same
BThey are 3/6 and 3/9, so they differ
CThey are 3/9 and 3/6, so they differ
DBoth equal 3/12, so they are the same
Answer & Solution
Correct answer: B. They are 3/6 and 3/9, so they differ
1. Conditioning on B reduces the sample space to the six numbers above 13.
2. Three of those six are even, so P(A given B) is 3/6.
3. Conditioning on A instead reduces the sample space to the nine even numbers.
4. Three of those nine exceed 13, so P(B given A) is 3/9.
5. The two conditionals therefore differ, because the reduced sample spaces are different sizes.
6. A value of 3/19 would be the joint probability, which conditions on nothing at all.
_Source: OpenStax Introductory Business Statistics (CC BY 4.0), Ch 3 "Probability Topics", section 3.1 Terminology_
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