**Problem 1.3.4: Probability in Sample Spaces**Given:- The sample space is $ C = C_1 \cup C_2 $.- The probability of event $ C_1 $ is $ P(C_1) = 0.8 $.- The probability of event $ C_2 $ is $ P(C_2) = 0.5 $.Task: - Find the probability of the intersection of events $ C_1 $ and $ C_2 $, denoted as $ P(C_1 \cap C_2) $. Solution:Using the principle of inclusion-exclusion, we can express the probability of the union of two events as:\[ P(C_1 \cup C_2) = P(C_1) + P(C_2) - P(C_1 \cap C_2) \]Since the sample space $ C $ is $ C_1 \cup C_2 $, we assume:\[ P(C_1 \cup C_2) = 1 \]Therefore:\[ 1 = 0.8 + 0.5 - P(C_1 \cap C_2) \]Solving for $ P(C_1 \cap C_2) $:\[ 1 = 1.3 - P(C_1 \cap C_2) \]\[ P(C_1 \cap C_2) = 1.3 - 1 = 0.3 \]Thus, the probability of the intersection $ P(C_1 \cap C_2) $ is $ 0.3 $.

Name: **Problem 1.3.4: Probability in Sample Spaces**Given:- The sample spa
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Description: **Problem 1.3.4: Probability in Sample Spaces**Given:- The sample space is $ C = C_1 \cup C_2 $.- The probability of event $ C_1 $ is $ P(C_1) = 0.8 $.- The probability of event $ C_2 $ is $ P(C_2) = 0.5 $.Task: - Find the probability of t

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Answered: 1.3.4. If the sample space is C = C1UC2…

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1.3.4. If the sample space is C = C1UC2 and if P(C) = 0.8 and P(Ca) =05 find

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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Concept explainers

Types of Bias

Statistics is the study of data collection, organization, analysis, interpretation, and presentation. Statistical bias is a characteristic of a statistical technique or its findings in which the expected value deviates from the actual root quantitative parameter being estimated. According to the actual definition of bias, it refers to the tendency of a statistic to overestimate or underestimate a parameter.

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Question

**Problem 1.3.4: Probability in Sample Spaces**

Given:
- The sample space is $ C = C_1 \cup C_2 $.
- The probability of event $ C_1 $ is $ P(C_1) = 0.8 $.
- The probability of event $ C_2 $ is $ P(C_2) = 0.5 $.

Task:
- Find the probability of the intersection of events $ C_1 $ and $ C_2 $, denoted as $ P(C_1 \cap C_2) $.

Solution:
Using the principle of inclusion-exclusion, we can express the probability of the union of two events as:
\[
P(C_1 \cup C_2) = P(C_1) + P(C_2) - P(C_1 \cap C_2)
\]

Since the sample space $ C $ is $ C_1 \cup C_2 $, we assume:
\[
P(C_1 \cup C_2) = 1
\]

Therefore:
\[
1 = 0.8 + 0.5 - P(C_1 \cap C_2)
\]

Solving for $ P(C_1 \cap C_2) $:
\[
1 = 1.3 - P(C_1 \cap C_2)
\]

\[
P(C_1 \cap C_2) = 1.3 - 1 = 0.3
\]

Thus, the probability of the intersection $ P(C_1 \cap C_2) $ is $ 0.3 $.

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