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Q13E

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Found in: Page 477

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# 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 $$p\left( {{F_3}} \right) = \frac{1}{2}$$

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

See the step by step solution

## 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

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

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

$$= \frac{3}{{17}}$$

$$= 0.176$$

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