   Chapter 9.9, Problem 9ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove Bayes’ theorem for n=2. That is, prove that if a sample space S is a union of mutually disjoint events B 1 and B 2 if A is an event in S with P ( A ) ≠ 0 , and k = 1 or k = 2 , then P ( B k | A ) = P ( A | B k ) P ( B k ) P ( A | B 1 ) P ( B 1 ) + P ( A | B 2 ) P ( B 2 ) .

To determine

To prove the Bayes’ theorem for n=2.

Explanation

Given information:

A sample space S is a union of mutually disjoint events B1 and B2, if A is an event in S with P(A)0 and if k=1 or k=2, then

P(Bk|A)=P(A|Bk)P(Bk)P(A|B1)P(B1)+P(A|B2)P(B2)

Calculation:

It is given that S is a union ofmutually disjoint events B1 and B2.

S=B1B2A=ASA=A(B1B2)A=(AB1)(AB2){since A is any event is S therefore (AB1) and (AB2) are mutually disjoint}P(A)=P(AB1)+P(AB2)P(A)=P(B1)P(A|B1)+P(B2)P(A|B2)

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