Discussion 6

Chapter 4 - DISCRETE CONDITIONAL PROBABILITY

Problem 26

Suppose that A and B are events such that \(P(A \mid B)\) = \(P(B \mid A)\) and \(P(A \cup B)\) = 1 and \(P(A \cap B)\) > 0. Prove that P(A) > 1/2.

From the first information:

\[\frac {P(A,B)}{P(A)} = \frac {P(A,B)}{P(B)} = P(A) = P(B)\]

The second part of the information provides that:

\[P(A \cup B) = P(A) + P(B) - P(A \cap B) = 1\]

Therefore:

\[2P(A) - P(A \cap B) = 1\] \[P(A) = \frac {1 + P(A \cap B)}{2} = P(A) > \frac {1}{2}\] Where;

\[P(A \cap B) > 0\]