MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Does this answer look like I calculated correctly. The question is in a certain population of women 4% have had breast CA, 20% are smokers, and 3% are smokers and have had breast CA. A woman is selected at random from the population. What is the probability that she has had breast CA or smokes or both.
### Explanation of Probability Problem Involving Breast Cancer and Smoking

#### Instructions
- **Problem Statement**: Do the problem #11 on page 78 in the Chapter 3 Review questions and exercises starting on page 76.

#### Diagram
The diagram is a Venn diagram used to visualize the probabilities of two events, A and B:
- **A** represents individuals with breast cancer.
- **B** represents individuals who are smokers.

The Venn diagram has two intersecting circles with the following data:
- **Circle A (Breast Cancer)**:
  - Breast Cancer only: 4% (0.04)
- **Circle B (Smokers)**:
  - Smokers only: 20% (0.20)
- **Intersection of A and B (Both Breast Cancer and Smokers)**:
  - Both: 3% (0.03)

#### Probability Calculation
The goal is to find the probability that a person has either breast cancer, smokes, or both.

The formula used is: 

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Where:
- \( P(A \cup B) \) is the probability of having either event A or event B or both.
- \( P(A) \) is the probability of having breast cancer: 0.04
- \( P(B) \) is the probability of being a smoker: 0.20
- \( P(A \cap B) \) is the probability of having both breast cancer and being a smoker: 0.03

The calculation follows these steps:
1. **Add probabilities of each individual event**:
    \[ P(A) + P(B) = 0.04 + 0.20 \]
2. **Subtract the intersection of both events**:
    \[ P(A \cup B) = 0.04 + 0.20 - 0.03 \]

This simplifies to:
\[ P(BC \cup Sm) = 0.24 - 0.03 = 0.21 \]

#### Conclusion
The probability that an individual has had breast cancer, smokes, or both is calculated to be 0.21 (21%).

This exercise demonstrates how to use the addition rule of probability involving overlapping sets (events) to avoid double-counting the intersection.
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Transcribed Image Text:### Explanation of Probability Problem Involving Breast Cancer and Smoking #### Instructions - **Problem Statement**: Do the problem #11 on page 78 in the Chapter 3 Review questions and exercises starting on page 76. #### Diagram The diagram is a Venn diagram used to visualize the probabilities of two events, A and B: - **A** represents individuals with breast cancer. - **B** represents individuals who are smokers. The Venn diagram has two intersecting circles with the following data: - **Circle A (Breast Cancer)**: - Breast Cancer only: 4% (0.04) - **Circle B (Smokers)**: - Smokers only: 20% (0.20) - **Intersection of A and B (Both Breast Cancer and Smokers)**: - Both: 3% (0.03) #### Probability Calculation The goal is to find the probability that a person has either breast cancer, smokes, or both. The formula used is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Where: - \( P(A \cup B) \) is the probability of having either event A or event B or both. - \( P(A) \) is the probability of having breast cancer: 0.04 - \( P(B) \) is the probability of being a smoker: 0.20 - \( P(A \cap B) \) is the probability of having both breast cancer and being a smoker: 0.03 The calculation follows these steps: 1. **Add probabilities of each individual event**: \[ P(A) + P(B) = 0.04 + 0.20 \] 2. **Subtract the intersection of both events**: \[ P(A \cup B) = 0.04 + 0.20 - 0.03 \] This simplifies to: \[ P(BC \cup Sm) = 0.24 - 0.03 = 0.21 \] #### Conclusion The probability that an individual has had breast cancer, smokes, or both is calculated to be 0.21 (21%). This exercise demonstrates how to use the addition rule of probability involving overlapping sets (events) to avoid double-counting the intersection.
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