**Probability and Independence of Events**In a probability space, consider events \(E\) and \(F\) such that the probabilities \( \mathbb{P}(E) \neq 0 \) and \( \mathbb{P}(F) \neq 0 \). Additionally, it is given that:\[ \mathbb{P}(E|F) = \mathbb{P}(F|E) \]The question is whether this implies that \(E\) and \(F\) must be independent. **Options:**- ○ true- ○ false

The given statement is false here. We find a counter example to prove that statement is false here.

Answered: If E and F are events in a probability…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Part 1: Suppose that F and X are events from a common sample space with P(F) # 0 and P(X) # 0. Prove that P(X) = P(X|F)P(F) + P(X|F)P(F). Hint: Explain why P(X|F)P(F) = P(Xn F) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1. %3D Explain why P(F|X) = P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem.

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If E and F are events in a probability space with P(E) + 0, P(F)+ 0, and P(EF) = P(F|E) then E and F must be independent. O true O false