Question

Question: Suppose that \(A\) and \(B\) are events from a sample space \(S\) such that \(p(A) \ne 0\) and \(p(B) \ne 0\). Show that if \(p(B\mid A) < p(B)\), then \(p(A\mid B) < p(A)\).

Short answer

Answer

The equation \(P(A/B) < P(A)\) has been proved

Step-by-step solution

Given data

The given equations are \(P(B/A) < P(B)\) and \(P(A) \ne 0,P(B) \ne 0\).

Concept of Probability

If\(S\)is a finite non-empty sample space of equally likely outcomes and\(E\)is an event which is a subset of\(S\).

The probability of V\(E\) is \(P(E) = \frac{{\left| E \right|}}{{\left| S \right|}}\).

Calculation to show that \(p(A\mid B) < p(A)\) 

Here, \(P(B/A) = \frac{{P(A \cap B)}}{{P(A)}} < P(B)\).

It gives, \(\frac{{P(A \cap B)}}{{P(B)}} < P(A)\).

But \(P(A/B) = \frac{{P(A \cap B)}}{{P(B)}}\) hence, it gives that \(P(A/B) < P(A)\).