A poll conducted by Pugh Research found 75% of Americans want the next president to be middle-aged (35-65 years old). 57% responded that they preferred a president willing to compromise on issues to accomplish action rather than make decisions strictly along partisan ideals. 48% of the respondents fell into both categories. a. What is the probability it takes four attempts to find the first American who wants the next president to compromise on issues? b. What is the likelihood it takes at least four selections to find the first American who belongs to neither category? c. What is the probability the first American who wants the next president to be middle-aged is between the second and fifth attempts?
A poll conducted by Pugh Research found 75% of Americans want the next president to be middle-aged (35-65 years old). 57% responded that they preferred a president willing to compromise on issues to accomplish action rather than make decisions strictly along partisan ideals. 48% of the respondents fell into both categories. a. What is the probability it takes four attempts to find the first American who wants the next president to compromise on issues? b. What is the likelihood it takes at least four selections to find the first American who belongs to neither category? c. What is the probability the first American who wants the next president to be middle-aged is between the second and fifth attempts?
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Transcribed Image Text:### Research Findings on Americans’ Preferences for the Next President
A poll conducted by Pugh Research found key insights into what Americans desire in their next president:
- **Middle-Aged Preference**: 75% of Americans want the next president to be middle-aged, specifically between 35-65 years old.
- **Compromise on Issues**: 57% of respondents expressed a preference for a president who is willing to compromise on issues to achieve practical outcomes rather than adhering strictly to partisan lines.
- **Both Categories**: 48% of respondents indicated they fell into both aforementioned categories.
### Questions and Probabilities
1. **Probability Analysis for Compromise on Issues**:
- **Question**: What is the probability it takes four attempts to find the first American who wants the next president to compromise on issues?
- **Explanation**: This involves calculating the probability using geometric distribution.
2. **Likelihood for Neither Category**:
- **Question**: What is the likelihood it takes at least four selections to find the first American who belongs to neither category?
- **Explanation**: This involves calculating the probability given that a respondent does not belong to either category, using complementary probabilities.
3. **Probability Analysis for Middle-Aged Preference**:
- **Question**: What is the probability the first American who wants the next president to be middle-aged is between the second and fifth attempts?
- **Explanation**: This involves summing the probabilities for each instance using geometric distribution principles, focusing on attempts 2 through 5.
Understanding these probabilities helps in assessing how prevalent certain preferences are within the population and can guide future polling and research efforts.
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