Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Question: Let E be the event that a randomly generated bit string of length three contains an odd number of 1s, and let F be the event that the string starts with 1. Are E and F Independent?

Answer

The two events E and F are Independent events as $P (E \cap F) = P (E) \times P (F)$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

E is the event that a randomly generated bit string of length three contains an odd number of 1s, and F is the event that the string starts with 1

Step 2: Definition of Independent Events

Independent events are those events whose occurrence is not dependent on any other event.

Two events are Independent if the equation $P (E \cap F) = P (E) \times P (F)$ holds true.

Step 1: Given Information

E is the event that a randomly generated bit string of length three contains an odd number of 1s, and F is the event that the string starts with 1.

Step 3: Calculating the Probability

We know that, E is the event that a randomly generated bit string of length three contains an odd number of 1s.

In general, a bit has a single binary value, either 0 or 1.

Here, totally four bit strings of length three that contains an odd of 1s; 100, 010, 001, 111.

So, the event E becomes,

$E = \{100, 010, 001, 111\} a n d |E| = 4]$

Consider F be the event that the string starts with ‘1’.

Here, totally four bit strings of length three that starts with 1; 100, 110, 101, 111.

So, the event F becomes,

$F = \{100, 110, 101, 111\} a n d |F| = 4$

Next we find $E \cap F$ as

$E \cap F = \{100, 010, 001, 111\} \cap \{100, 110, 101, 111\} = \{100, 111\} a n d, |E \cap F| = 2$

In general, the sample space S of an experiment is the set of all possible outcomes.

In this experiment the set of all possible outcomes are 8.

Hence, they are, $\{000, 010, 001, 100, 110, 011, 101, 111\}$

So, $|S| = 8$

Step 4: Determining whether the events E and F are Independent or not

Now, we determine that the events E and F are Independent or not, from the definition of Probability.

Suppose that E is an event that is a subset of sample space S, then the probability of an event E is defined as,

$P (E) = \frac{|E|}{|S|} - - - - - - - (1)$ Where, $|E|$ is the number of outcomes in E, $|S|$ is the total number of outcomes.

Now, from the definition of Independent sets, Let E and F are two events, then the events E and F are Independent only and only if

$P (E \cap S) = P |E| \times P |S| - - - - - - (2)$

Using (1) we find $P |E|$ and $P |F|$ as,

$P (E) = \frac{|E|}{|S|} = \frac{4}{8} = \frac{1}{2} a n d P (F) = \frac{|F|}{|S|} = \frac{4}{8} = \frac{1}{2}$

Next we find

$P (E \cap F) = \frac{|E \cap F|}{|S|} = \frac{2}{8} = \frac{1}{4}$

We can write this as,

$P (E \cap F) = \frac{1}{4} = \frac{1}{2 \times 2} = \frac{1}{2} \times \frac{1}{2} = P (E) \times P (F)$

Hence, $P (E \cap F) = P (E) \times P (F)$ .

Therefore, the two events E and F are Independent.

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Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Question: Let E be the event that a randomly generated bit string of length three contains an odd number of 1s, and let F be the event that the string starts with 1. Are E and F Independent?

Step by Step Solution

Step 1: Given Information

Step 2: Definition of Independent Events

Step 1: Given Information

Step 3: Calculating the Probability

Step 4: Determining whether the events E and F are Independent or not

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Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Question: Let E be the event that a randomly generated bit string of length three contains an odd number of 1s, and let F be the event that the string starts with 1. Are E and F Independent?

Step by Step Solution

Step 1: Given Information

Step 2: Definition of Independent Events

Step 1: Given Information

Step 3: Calculating the Probability

Step 4: Determining whether the events E and F are Independent or not

Most popular questions for Math Textbooks

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