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Q13E

Expert-verified

Discrete Mathematics and its Applications

Found in: Page 477

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Question: Suppose that \(E, {F_1},{F_2}\,and {F_3}\)are events from a sample space S and that \({F_1},{F_2}\,and {F_3}\) are pair wise disjoint and their union is S. Find \(p\left( {\frac{{{F_1}}}{E}} \right)\)if \(p\left( {\frac{E}{{{F_1}}}} \right) = \frac{1}{8},p\left( {\frac{E}{{{F_2}}}} \right) = \frac{1}{4},p\left( {\frac{E}{{{F_3}}}} \right) = \frac{1}{6},p\left( {{F_1}} \right) = \frac{1}{4},p\left( {{F_2}} \right) = \frac{1}{4}\) and \(p\left( {{F_3}} \right) = \frac{1}{2}\)

Answer:

\(p\left( {\frac{{{F_1}}}{E}} \right) = 0.176\)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

\(p\left( {\frac{E}{{{F_1}}}} \right) = \frac{1}{8},p\left( {\frac{E}{{{F_2}}}} \right) = \frac{1}{4},p\left( {\frac{E}{{{F_3}}}} \right) = \frac{1}{6},p\left( {{F_1}} \right) = \frac{1}{4},p\left( {{F_2}} \right) = \frac{1}{4}\)and \(p\left( {{F_3}} \right) = \frac{1}{2}\)

Step 2: Formula used

\(p\left( {\frac{E}{{{E_1}}}} \right) = \frac{{p\left( {\frac{{{E_1}}}{E}} \right)p\left( E \right)}}{{p\left( {\frac{{{E_1}}}{E}} \right)p\left( E \right) + p\left( {\frac{{{E_2}}}{F}} \right)p\left( F \right)}}\)

Step 3: Calculating

By the generalized version of Bayes’ theorem

\(p\left( {\frac{{{F_1}}}{E}} \right) = \frac{{p\left( {\frac{E}{{{F_1}}}} \right)p\left( {{F_1}} \right)}}{{p\left( {\frac{E}{{{F_1}}}} \right)p\left( {{F_1}} \right) + p\left( {\frac{E}{{{F_2}}}} \right)p\left( {{F_2}} \right) + p\left( {\frac{E}{{{F_3}}}} \right)p\left( {{F_3}} \right)}}\)

\( = \frac{{\left( {\frac{1}{8}} \right)\left( {\frac{1}{4}} \right)}}{{\left( {\frac{1}{8}} \right)\left( {\frac{1}{4}} \right) + \left( {\frac{1}{4}} \right)\left( {\frac{1}{4}} \right) + \left( {\frac{1}{6}} \right)\left( {\frac{1}{2}} \right)}}\)

\( = \frac{3}{{17}}\)

\( = 0.176\)

\(p\left( {\frac{{{F_1}}}{E}} \right) = 0.176\)

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Question: Suppose that \(E, {F_1},{F_2}\,and {F_3}\)are events from a sample space S and that \({F_1},{F_2}\,and {F_3}\) are pair wise disjoint and their union is S. Find \(p\left( {\frac{{{F_2}}}{E}} \right)\)if \(p\left( {\frac{E}{{{F_1}}}} \right) = \frac{2}{7},p\left( {\frac{E}{{{F_2}}}} \right) = \frac{3}{8},p\left( {\frac{E}{{{F_3}}}} \right) = \frac{1}{2},p\left( {{F_1}} \right) = \frac{1}{6},p\left( {{F_2}} \right) = \frac{1}{2}\) and \(p\left( {{F_3}} \right) = \frac{1}{3}\)

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