Understand conditional probability and connect independence to unchanged conditional probabilities.
Calculate and compare two reversed conditional probabilities
Problem
With overlap \(12\), \(A\) total \(30\), and \(B\) total \(20\), calculate \(P(A|B)=12/20\) and \(P(B|A)=12/30\), reduce \(\text{both}\), and compare them.
Big Picture
What this problem is really about
Reversing a conditional keeps the overlap numerator but changes the conditioned-group denominator. We’ll form each ratio from its own condition, reduce both fractions, compare them on a common scale, and use the difference to show why conditional probability is not symmetric.
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